I felt bad about my nervousness in interview situations too, so I started taking improv classes. Now interviews don't give me any performance anxiety at all; once you've performed in front of a large audience with no idea what you're going to say or do, it makes interviews a cakewalk.
And remember, even though the primary skill you're interviewing for is coding, they are also testing you for communications ability. Being able to clearly and concisely explain what you're doing is at least as important as being able to get things done.
I've been doing improv for 10+ years and had never realised this, but it may well be a large part of the reason why I don't get nervous in interviews. Thanks for helping me make the connection!
Plus, as one other commenter points out elsewhere in this thread, once you've been on the other side of the table for a while, you get a much better idea of where your own strengths lie (and what you're up against, which can instil a lot more confidence than you might think!).
And remember, even though the primary skill you're interviewing for is coding, they are also testing you for communications ability. Being able to clearly and concisely explain what you're doing is at least as important as being able to get things done.