You'll have to stick to your guns and wait it out. If you give in, you'll prolong it, but staying strong will help her get stronger too. Don't make excuses like, "she's just sensitive, or just competitive" because to outsiders that reads: "I am delusional about my spoiled kid." You'll get a more compassionate response with "We're trying to redirect her and help her develop coping skills. It's a work in progress." Then she won't be perceived as spoiled, or you as negligent.
My sister has 3 kids, the first was a peach, the second a screamer, and the third, a peach. The screamer was at odds with a little sister 2 1/2 years younger, but her screamer personality was evident long before that. And just like your daughter, would scream for apparently nothing, but in retrospect, it was probably just over stimulation. Errands, too many lights on, too much sun, babies crying, any of it. Time outs didn't work. They probably just made her feel nervous or misunderstood and worsened the screaming. My sister stuck to the discipline, and repeated herself over and over again with the rules, and now she has a very type-A but very poised 10 year old girl. I think your daughter would have been a screamer even without the twin sisters.
I had 2 peaches and then a screamer. Every time I was around a little screamer, I had the "OMG, those parents need to stop giving into that child" reaction. That's not entirely the case. Kids have different levels of coping skills and different coping styles, just like adults. My first two coped by being introverted and thoughtful, which looks like good manners to an outsider. This is worse than screaming because you never know when something is wrong, so you have to remember to ask them how they feel all of the time, even when things seem to be going along fine. My screamer reacts to stress, fatigue, change, hunger, sadness, too much sun, not enough sun, by screaming. At least screamers make that part easier for us, because we know there's something up, just need to do a little detective work.
As parents, we always acknowledge our kids feelings and it turns into this big kid centered world. What has been working for me is to ask them a series of questions about how they feel, and also about how they think other people feel in the same situation. My little 4 yr old screamer has made a lot of progress because he is beginning to mentally play back those questions, i.e. brother can't play with him because he has to leave for practice, that would have set off a screaming fit and people would say, "how sweet, he loves his brother." and then I would say, "no, that's really manipulative, to use feelings as an excuse for bad behavior." So I would sit him down and ask him a series of yes questions,{{ "do you love your brother? Do you want him to be happy and successful? Is it good for your brother to go to practice? Does his coach deserve respect? Should your brother show respect by showing up on time? Can you show your brother that you love him by helping him get to practice on time? Ok...go grab a water bottle, glove, etc.}} Now he's in action to help his brother go instead of crying to get him to stay.
I learned this from my sister, and my screamer is gaining coping skills faster than her screamer did, thank god I had someone with experience to learn from. My sister had to suffer through trial and error. It's still a long slow process though.