I think a lot of us worry too much about what others may think of us. I also want to be a good rider, and am terrified of someone looking at me and saying I'm terrible, they feel sorry for my horse. Not when I ride at my home barn, but in front of strangers. To help get over that I just ask myself who that person is to me - nobody. Why does their opinion matter - it doesn't. If my ride doesn't go well today what will that mean in a week/month/year - nothing.
My trainer has always told us to laugh or sing when we get nervous and it does help.
I have to force myself to ride through my fears, which are always unfounded. It sometimes takes me 15 minutes to actually mount up. I've discovered that when things do happen, and I survive them, it actually makes me bolder in the moment, but then I think about it before the next ride and get anxious. It's horrible how we psych ourselves out.