There are several things you should ask yourself if you catch your significant other looking through your texts. One might be, why would they go through them? Have you been acting suspicious? Are you flirtatious? Do you get texts at odd hours in front of him? Are you, in fact, cheating? So many friends of mine have had a suspicion that their boyfriend was cheating on them, and the boyfriend would deny it to no end, to the point where they would get mad at their girlfriends for questioning their loyalty. If those girls hadn’t gone through their guy’s texts (or in some more serious cases, e-mails), they probably would have never known that their guy was cheating on them. I think that if you really suspect that your boyfriend is fooling around, then you should either dump him or do some investigating. Sorry, that’s just how I feel. I was in a relationship where the guy cheated, and trust me, if you feel like your guy is cheating, then he probably is.
Okay, so now that that part’s done with, another question you might want to ask yourself is whether or not you think your significant other is a creeper. If they’re going through phone texts in the dark by a candle after they’ve watched you fall asleep and are by the phone going "good, good" like an evil maniac, then they’re probably a creeper. Dump ‘em.
Now, if neither one of these cases apply, maybe your significant other has just been hurt before. Maybe they really like you and are just afraid of you cheating on them or hurting them like they were hurt before. It may not necessarily be that they don’t trust you, but that they just don’t trust people in general. If you really like them, I would say try to make it work and show them that they can trust you. I won’t judge you though if you decide it’s not worth it. You’re young after all and I don’t want to go all Oprah on you. Just try to put yourself in their shoes and honestly do what you feel is right for your relationship.