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She has you framed in her mind a certain way, and even though you were telling the truth, she will not believe you. Some people have an erroneous belief that "a leopard doesn't change his spots" and they actually believe others cannot change. This seems to be happening here.
I've been in these sorts of toxic relationships in the past. Drop the girl and all the fucking drama is what I would do. She doesn't trust you. Find a girl who does.
I wouldn't call myself an expert by any means, but I'll chime in here with my thoughts anyway.
I second what Oceanx said in the way that when trust is broken, it can be nigh impossible to reestablish... And in all fairness, you are only paying the price of your own actions. Shouldn't have checked her phone in the first place.
ANYWAY, I think that straight up dropping this relationship is a bit radical if you really care about it. If you want to stay in it, I guess you will have to try to reestablish the trust that has been lost. The way I would go on about it, if I was you, is actually openly discussing the issue. Tell your girl that yes, you made a mistake in the past and you accept the blame for it. Also convey that you have improved and would never do such a thing again, but that you need her to make a conscious effort as well and to stop blaming you at the slightest chance, since this is seriously damaging your relationship and you have absolutely no way to counteract it. Once this is done, this is where the "tough" part begins... Just don't do this ever again. Don't give her a reason to distrust you.
Please note that this will only work if she also puts in some effort from her side, and you should convey that as well. If you absolutely cannot fix those trust issues, then, well... The relationhip is really fucked and you should start thinking about your way ouy.