How to get rid of social anxiety?

practice and more practice should do the trick... you probably wont get rid of it but for sure it will make it better

Some people aren’t comfortable with public displays of affection. That’s not social anxiety.
Tell your girlfriend you aren’t comfortable with it, and repeat yourself as necessary. That way she can avoid taking your discomfort personally.
Be affectionate with her in private.

It's not necessarily "social anxiety" if you'd rather not hang all over each other in public. Some people just don't care for PDA. Some people don't feel the need to hang all over their significant other, whether in public or private. I personally don't want to see a couple making out or fawning all over each other when I'm trying to eat my lunch or enjoy a concert or ride the subway to work. There's no need for them to show off like that.

It's not necessarily a bad thing to be concerned about what other people think of you. It's one thing if this anxiety is keeping you from something you really want to do (assuming it's nothing harmful or dangerous). But it's really not an awful thing to think, "Gee, if I do XYZ in public people are going to be grossed out/find me obnoxious/think of me as an attention hog." We live in a society, and while we all have freedom we all also have a responsibility to act right and to consider other people's feelings and comfort.

Bottom line, though ... you need to talk to your GF about this. "I hope you know by now that I care for you very deeply. I'm not into PDA and I would rather keep our physical relationship just between the two of us. I'm not embarrassed of you or hiding anything."

You can offer a compromise, like a kiss here and there, or holding hands walking down the street. But if you don't want to be hugging and making out during a concert (like many obnoxious couples I've seen), tell her so - "I'm really not comfortable with that. I've told you how I feel about you, and I don't think we need constant physical contact to prove that."

If you cannot reach a compromise about this, then it may be time to go your separate ways: "Look, Shelly, I told you I care about you very much. But if you need constant physical attention in order to believe that, then maybe we're not meant to be together because I don't feel that that's necessary. Maybe you'd be happier finding someone else who thinks the same way as you about this."