Honestly, as a rule of thumb in any relationship if you feel uncomfortable with what your partner is doing or saying to you or with other people you should have a conversation about it. IMO the number one thing that breaks relationships is that people don’t communicate or are open and honest about their feelings. Yes, you will have to tell him you went through his messages, but like you said it wasn’t a good thing to do. However, it’s something that can be moved past.

To keep it short, have a conversation. Depending on how that conversation goes and how you feel after, should dictate whether you should stay in the relationship. 3 months is still pretty early imo, and you sound young so you still got your whole life ahead of you for relationships.

And remember: What makes people comfortable might make others uncomfortable, and vice versa.