This is perfectly normal, though unpleasant.
I agree with Giuseppe Biondi-Zoccai, whispering or talking very quietly is a better choice. If you yell all the time, the child will stop hearing it as an imperative. It becomes 'normal'.
As parents we have to decide what is important and what is not. It is very easy to become bossy. We do not want to take choices away from children, it causes all sorts of disciplinary problems down the road. If it doesn't matter which shirt they wear, let them choose. If it does matter enough to fight over, then you cannot back down once you've made that choice.
Behaviourists will tell you that if it takes 15 minutes of tantrum for a child to get what they want, we have taught them to keep screaming. Next time they'll increase the time and volume until the parent 'gets the message'. This is a great time to teach, 'no means no'. If you think it over and discover that you were wrong or unfair, then you admit you made a mistake -- but never during a tantrum. This is why it pays to be a thoughtful parent. Breathe, think and then react.
Admitting to mistakes makes us better parents, by the way. We do not lose face by admitting that we know when we are wrong. We lose credibility when we never admit our errors.
Now stopping a tantrum is difficult, especially once the habit has started. I will assume that re-direction has not worked, but it should be what you try first and as soon as possible. You know the triggers. So my advice is 1) Pick your battles carefully. Saying yes or no today doesn't mean you always have to make that same choice. 2) Talk quietly and try to get on with your business and within reason, ignore the behaviour. If they know how bothered you are, they may continue with that as their goal. 3) Time out is not punitive, or should not be. It is a breathing space, so putting a tantruming child in time-out is counter-productive, That doesn't mean you can't ask an older child to go to their room, but at 15 months, you have to parent and make sure they are safe. I used to cook or wash the floors, clean the bathrooms, anything busy. I had to do those chores regardless and it meant that I could show my kid that life was continuing in spite of her tantrum. 4) Turn off other noise. The TV, the music and ask others to leave the area. During a tantrum less audience and less noise is the best choice. If there is another parent or caregiver, feel free to change places. Agree to handle 15 minutes and then it is the other adult's turn to calm down. If you are a single parent, it is so hard but you have to see it through.
Nearly every parent has been through this to one degree or another. It is very hard and extremely aggravating. So, it's a big club that no one wants to be a member of. Knowing that others have been through it probably doesn't help you much! Best of luck.
on edit: This comment would depend on the maturity level of your 15 month old. In our home TV, and electronics are earned. We do what we need to do for ourselves and our family and then we can fill our free time with the things we like to do. This doesn't mean we work all day without breaks, but that we have certain things we do before those breaks occur. TV is a privilege, and not a right. I think your daughter could possibly understand that she has lost or earned TV by her own actions.
If you used a token system, it might work. Decide how many tokens are needed for 1/2 hour of her show -- I suggest three because even toddlers can recognise that number even if they cannot count. Make three TVs and colour them -- display them where she can see them. If she has three TVs, she can watch a show. If she doesn't have three, she can't. If she yells or acts inappropriately, you remove one TV. Now she can't watch until she has improved her behaviour. If the tantrum continues (this will in the first few days make it worse), then after ten minutes remove another TV token. If you run out of TVs, it will take her at least thirty minutes of being good to earn her three TV tokens. When she is being good, you add a TV token and smile and tell her you noticed how well she is behaving.
You decide how much TV your child watches per day. If you allow an hour -- then make 3 green tokens and 3 red tokens. This show uses red tokens, that show uses green. You are the parent and you do control this aspect of your child's life.
This is an overview. I am not supplying you with a management program but with an idea. Though I did do this for a living, this is the internet. I have not met you or what is even more important, I have not met your child.