Is it normal for this?

sorry but you have a fake ass relationship he is a cheater in till you realize this your eyes will be closed and till it is to late happen to me 7 years ago

Sounds like he is considering cheating you. He pretends you two are not in a relationship in public means he want's other women to spot his "availability".

You emberrass him. That's why.

Does he tell other people that you guys are dating? Does anyone else know that you two are dating? Is he trying to hide it from other people? If any of that is the case, no, it's not normal and he is a jerk and you deserve a lot better than that. If that's not the case, he might just have PDA issues and you should talk to him about that.

That's dumb. Maybe he's not a fan of PDA.

Totally normal, he's just shy. Well, he might be. I don't know. But wouldn't it be better this way? Imagine that you guys are all lovey-dovey and some girls start liking him (you know that "want what you don't have")...Wouldn't you want him to show his feelings to you and to you only?

Could be that he's embarrassed to be your boyfriend. And honestly, I think that there is a part of you that knows this but doesn't want to admit it. I suggest talking to him.