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?
I'm sure many people feel the same way so you are not alone, especially people who are shy or introverted.
I'm not shy, but definitely introverted.  I have had issues with self-consciousness since I was in junior high school
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