I often feel self-conscious when I'm out in public
That is an unpleasant feeling and you feel like people are watching you
Exactly.  I don't know what to do to prevent it
feelings of self-consciousness are sometimes associated with shyness or paranoia.
Impairment.
When feeling self-conscious, one becomes aware of even the smallest of one's own actions. Such awareness can impair one's ability to perform complex actions. 
Adolescence is believed to be a time of heightened self-consciousness. A person with a chronic tendency toward self-consciousness may be shy or introverted.
Psychology.
Unlike self-awareness, which in a philosophical context is being conscious of oneself as an individual
You don't know how you look on stage, but if what you're doing reads clearly it means you have to try to be quite honest, because otherwise it shows. You have to be very aware of how you work with yourself - it all comes from what you feel at that moment. When I do James, for instance, I actually say the lines to myself as I'm doing the mime. Not always, but... you have to have the overall feeling at that moment, in that scene;