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
It's very difficult and maybe therapy or medication. That kind of awareness can impair your ability to perform certain actions.
I wonder how many people it happens to regularly?
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
I [didn't] want to screw up my lines." She further relayed, "There are many times when I think I didn't get a scene. When you don't feel anything, it's frustrating. I'm less hard on myself because I realize how difficult what we do is. But, as an actor, you hope that you go in there and get caught up in the moment. I think, self-indulgently, that's why we become actors in the first place — to feel