First, do not make the mistake of assuming that this topic is simple, or that it is cut-and-dried. Also, I apologize in advance for this answer being SO long; maybe take it as an illustration of the complexity of the topic. AND because you've said she has a hard time explaining WHY she thinks/feels the way she does, I thought I might go a little in-depth with some possibilities for considerations that might be motivating her, so you can listen to her from some perspective other than "she's wrong and I have to change this."
It does sound like the idea of racial superiority that this girl has is coming from her family/other intimate acquaintances, and as someone mentioned already, this means that attacking that belief might be more of an emotional battle than a logical one, and could easily alienate her. It often takes a very long time (into mid 20's or so) for a young person to critically examine the values they were raised with enough that they have the understanding and confidence to replace those values with something else. It often requires a paradigm shift or epiphany moment to actually change the belief on the inside rather than just changing what we preach because we come to understand that that's what's expected, what we are told we should believe.
You should remember that you aren't looking to change what she says about this topic, you're looking to teach her to think and feel differently about it. Which means identifying the underlying conversation, assessing its validity, and replacing it with something better that she finds understandable and agreeable - if validity is found wanting in the current underlying conversation. That would be my second bit of advice: just because the surface conclusion (racial superiority) is erroneous, doesn't mean that the underlying conversation doesn't have some merit. It might. Be prepared for that.
I was thoroughly indoctrinated during my upbringing in the belief that all people are equal, and that nobody is superior to anyone else. That was what I was told to believe, by parents and school and peers etc. I was taught that if people behaved badly, it was because they had hard lives and didn't know how else to react. I was taught that if some people were successful and others weren't, it was because the successful ones had had lucky breaks which the unsuccessful had not been privy to. This conversation is separate from, but parallel to the conversation about race, and gets wrapped up with the idea of race in some people's minds (and is therefore relevant to understanding feelings about race). I'll switch over to race here, and then come back to the flaws in the beliefs outlined in this paragraph in a second.
Before we studied Martin Luther King Jr in school, I was race-blind. There wasn't a reason to see children of other colors as different from myself. However, studying the Civil Rights movement gave me the idea that people can be distinguished on the basis of race. This assumes that people of different races are different. Else there would be no reason for that distinction. There is no superiority complex built into the ability to distinguish between things: just because I can say this apple is a gala and this is a Granny Smith doesn't indicate any elevation of one over the other. However, the superiority complex is built-in to our society in another, more insidious way. This is where the idea of "lucky breaks" mentioned in the above paragraph comes in:
We have affirmative action laws.
I'm not going to make any statement about whether these laws are necessary or effective (that's a completely separate discussion). However, I will make the claim that inherent in the statement that women and minorities need a leg-up in life is the assumption that being born a woman or minority is a disadvantage that must be overcome. If you're playing a game and given a handicap, the assumption is that you need that handicap to make the game fair, because the other player is measurably superior. And right there, built-in to our legal system, we have the conversation that being born white and/or male is a lucky break, and therefore a measure of superiority based purely on inborn biological characteristics.
Going into my early 20's, even though I thoroughly believed in my own mind that I believed all people of all races/backgrounds to be equal, I still felt on a gut level that I was better by virtue of being white, and I felt some small measure of pity for those who were not. And here's the kicker - i didn't even realize it. I was cured, though, by one of my college professors, with a simple lecture on the history of colonization and slavery, and while I don't remember all the sordid details, it went something like this:
When the countries of Western Europe started colonizing the rest of the world, they procreated with all of the indigenous peoples they came across, and this continued for generations with racial mixing occurring an uncountable number of times. In cultures that kept slaves (such as here in the US), it was almost common practice for slave owners to procreate with their slaves, so large-scale black and white racial mixing has been happening ever since Western Europe started trading African slaves in the 1400's. Therefore today a white person and a black person in the US are virtually indistinguishable on the basis of their DNA - meaning that if we are going back to the gala vs Granny Smith apple metaphor, there really is no meaningful difference, and no basis for distinction.
What was stunning to me about this lecture was not so much the idea that there is little to no discernible difference genetically between white people and black people in the US, but rather the fact that I could feel that that mattered to me. I could feel the way I regarded non-white people living in the US changing with this proclamation that we are basically the same. This is what made me aware that I was carrying a racial prejudice. After all, why should it matter to me? I am not that genetically similar, for example, to people of African decent who are still living in Africa. I am not that genetically similar to people living in East Asia. Does that mean I see those people as inferior?
No. But it does mean I see them as different. And they are, but not by virtue of race. Rather, by virtue of what, culturally, they have been raised to think and believe, and the kinds of things they have been taught to value - and also sometimes the kinds of skills they have acquired. The problem, then, is when people mistakenly associate a race with a negative behavioral pattern, or with certain harmful values or beliefs.
Because the inescapable truth is that people are different, and different, by definition, is not equal. No one is going to choose me over my brother for a task involving chemistry, and no one is going to choose my brother over me to give a cello performance. These are differences in capability based on choices we've made. Men cannot bear children, and the fastest women cannot run as fast as the fastest men. These are differences in capability based on physiology. I choose some people to be my friends and others not to be, and this choice is based on my judgments of those people's characters - their morals, personalities, and values. In other words, it is necessary to be discriminating in how we allocate our time, energy and resources, and this requires judging the value and merits of the people we encounter. And it would be silly in the extreme to say that "all people are equal and therefore must be judged to have equal merit." What, then, do we say about serial rapists and murderers? Or for that matter about college applicants? Or about reputation on stack exchange? If two patients simultaneously go into cardiac arrest and there is only adequate staff to treat one of them, the doctor will choose one of them to treat. How could that be possible if the two patients were truly equal?
Because "equal" needs a qualification: all people should be equal under the law. I say "should be" rather than "are," precisely because of laws which distinguish between different classes of people based on race, gender, etc.
This is a very long illustration of a very simple point: it is not necessarily wrong to regard some people as superior to others.
However, this judgement should be made not on the basis of race or gender or native language, or any other such factor. It should be made (mostly situationally) based on choices, and behaviors, and capabilities. And it should be noted that the second and third of these largely follow from the first. Someone who chooses to work hard, who treats people kindly, and who is an expert in their field, can reasonably be thought superior to someone who is always looking for a handout, is cruel to others, and has not cared to learn to be of value to his fellow man through the acquisition of any particularly useful skills. That may sound callous, but it is a fact.
I'm mentioning this because it could be that your 15-year-old charge had started to observe that some people are better than others and therefore the idea that all people are equal is a fallacy (possibly because some people are mistreating her? That was one of the early signals for me). Maybe race is the explanation she has been conveniently given to explain this clash, and she needs a better explanation. Namely, that there are good and bad eggs in every bunch. It's an individual thing, not (usually) a group thing. (Unless people choose to group themselves by a negative characteristic - aka KKK and Nazism - which then is a choice, and not an in-born characteristic).
I will also point out that there is serious backlash going on right now against the idea of racial guilt. The idea that people who have white ancestry owe reparations to people of other races because of the exploitive behaviors of their ancestors. White people are tired of being told they are guilty of some kind of racial crime just because they were born white. So if someone is saying they are proud to be white, it may be as simple as saying, "I refuse to feel guilty for a crime that I have never committed, or thought to commit, and that I have nothing to do with by virtue of being several generations removed from it." I would say the opposite of what I saw claimed in another answer (or possibly a comment): white racial identity is more about asking to be treated as people, rather than as white. Which I think is what it's all about for other racial groups as well - or at least I think it's safe to say it should be about that, though sometimes it seems like it's more about being treated as a group for the purpose of gaining access to entitlements.
One last thought: it seems that people who have more exposure to racial minorities have less fear of them and therefore less discrimination against them, possibly because they are then able to see how similar we are for themselves. In terms of human psychology, we are more able to trust those with whom we share (or at least think we share) values, so getting the chance to see firsthand that people by and large care about being good and valuable people can go a long way towards overcoming this. Maybe try getting her on a volley ball team in a more mixed neighborhood or something like that, maybe as a follow-up to talking about it.