My daughter is 4. When she had a tantrum, she sometimes scratches or hits herself. One morning she scratched too hard and left a mark on her forehead. I can tell clearly that she was embarrassed by this. And I don't know how to react when my friends started asking me in front of her, what happened to her forehead. I don't want to embarrassed my child further by telling the truth, but not telling the truth will send a message to her that it's alright to lie.
If you aren't sure how much she minds your telling, ask her. "Do you want me to say or would you rather pass?" Teach her how to say, "oh it was an accident" and that she has the right to not share any more than that if she doesn't want to.
This is one of those moments you could treat her like you would a fellow adult that wasn't in the room. If someone asked about a potentially embarrassing story about an adult and beloved friend or family member and knew you had all the gossip. Would you automatically share all the info? Just say, "Well she had a little mis-hap and got hurt. She'd rather not share more than that, so I want to respect her wishes." It's honest and respectful toward your child. It will also teach her that she can politely not answer prying questions later on in life if she wishes not to.
If it is helpful, there are also a few questions about tantrum avoidance on ParentingSE whose answers might help in avoiding similar mis-haps for your child in the future. The first question is about alternatives to spanking. I do not assume you are spanking your child or don't already have some ideas about how to deal with the tantrums themselves, but you might find some new and helpful ideas here.
This question is actually about what a parent can do in their home pre-two's to prevent some tantrums if not most. The answers generally give ideas for preventitive measures or attitudes toward tantrums once they start that help to remove or alleviate the behavior altogether. The chosen answer by Meg Coates is particularly good.