My girlfriend peeks at my cell phone text messages and phone records at any time. Can I marry such a person?

Can't! Love is based on mutual trust, and this "anytime" is absolutely abnormal. I was so suspicious of you before I got married, but I can imagine how troublesome my married life is. You can communicate with each other for her and for you. Breaking up and suspicion will never solve the problem, and it is likely to hurt both sides. I suggest you open her heart through communication and wish you success! !