There are three sides to this issue — his, hers and ours. They need to be worked on simultaneously, not one by one.
Yours: You mentioned that you get angry easily but not with her. That suggests to me that you have more control over your anger than you think you do. One way to prove to your girlfriend that you won’t “lose it” with her is to not lose it with other people either. There is nothing wrong with being angry. There is lots wrong with expressing it in a way that frightens other people, especially people you care about. Your job is to learn ways to express your anger in ways that are helpful, respectful and effective. If you don’t understand what that entails, do some internet search and/or get some coaching from a therapist. You can reassure your girlfriend a lot by demonstrating that you are working on it.
The two of you as a couple: Every couple goes through rough patches of misunderstandings and differences of opinion. The best way to avoid hurting each other is to develop skills in conflict resolution. Conflict doesn’t have to mean a fight. Couples that last are couples that learn how to be on the same team, fighting a problem rather than on different teams fighting each other.
Hers: She has apparently been very hurt in the past so her trust is low. She may be hypersensitive to anything that even vaguely looks like you are losing control. That’s something she can work on in therapy. Unfortunately, trust is something that is very hard to win. It’s up to her to decide whether she knows you well enough at this point to give you the gift of her trust. Yes, the gift. When we love someone, trust is the starting point. It’s hard to win it, which is why it ultimately is a gift. But it’s easy to lose it. Over time, she’ll learn that you aren’t going to throw away something as precious her trust.
I wish you well.