When you're first learning to play jazz piano, it can be pretty nerve-wracking to improvise in public. When you're playing written music, at least you know that the music is pretty good on its own; if you manage to play the right notes you'll sound OK. But with improv, it's different. No guarantees. You're in the "hot seat" and you're not sure how it's going to turn out.
Leaving aside for a moment the fact that you probably sound a lot better than you think, I've found that it's most effective to "ease into" playing jazz piano in public. Don't learn a bunch of jazz tunes and then commit to playing a full 4-hour gig. Instead, look for opportunities to play only a tune or two. Or better yet, work a little jazz improv into another musical situation.
A student of mine emailed me about this today. (Hi Mike!) He was involved in a concert where some children were in the audience, and he decided to play a kid's song for them. But instead of playing it the usual way (in this case, as a ballad), he put a light bossa nova beat under it and did some jazz soloing in the middle, utilizing his recently-learned A and B rootless voicings in his left hand.
He had a lot of fun as a performer, the kids loved hearing a fresh version of a familiar tune, and best of all for his development as a jazz pianist, he became more confident in his ability to successfully improvise in public. Bravo!!!
You can do this too. Look for small ways to improvise in public, without having to sustain long solos or play many tunes in a row. Build up your confidence and get some experience. (My first solo jazz performance was an 8-minute set at my high school talent show.) Then, little by little, find situations in which you can play for longer periods of time. This way, your jazz piano ability and your musical confidence will grow hand-in-hand.
Good luck and remember to have fun at every step of the way!
Check out my jazz piano video course if you want to become a fluent and confident jazz pianist. (I'll give you personal guidance and step-by-step instruction to teach you anything you want to learn.)