Trust Issue...

Me and my girlfriend have been going out for about 5 months now. We've been pretty happy but about a month into our relationship some other girl had came up to me trying to flirt, at first i was just trying to ignore her but she was playing dumb trying to hold my hand and hug me. I finally got her to back away from me but a friend of my girlfriend had seen me and got the wrong picture, thinking i was messing around with that other girl. My girlfriend heard and finally she believed me. But ever since, it has bothered her that if a girl were to try to push me to do something i wouldn't do anything to prevent something from happening. I know I wouldn't do anything like that. She tells me she can't trust me at all and I want to show her she can but I just don't know how. I've told her before that I can but i guess I just don't know how. She wants me to change quickly. I know i can be trustworthy but how can I show her the same??

This is more of her problem than yours. You didn't really do anything wrong. You didn't lie, didn't cheat, didn't act shady at all.

The best thing you can do is remain consistent and don't feed her drama. You should be gentle about her insecurity, but that's not the same thing as feeding the beast. She'll come to trust you if she really wants to.

How does she want you to change?

That threw large red flags in my mind. If you did nothing wrong, and she's baiting you to change because of something you didn't do - I'd be really concerned.