Like someone else said before, you can get injured doing absolutley anything, but yes I know its easier saying that then it is overcoming your fears.
I know its hard, but I think that the easiest way of overcoming somehting like that is, one self talk! Like you said your safe round horses and everything, so there isnt any reason there but freak accidents happen, so if you ever feel unconfident, you gotta talk to your self no matter how silly you look. Its reasurring yourself and your horse.
It doesnt always work, but try making your focus go on to something else if you fear something, or just think about how much fun you have with the horses.
Hope I helped
When you mix training and dedication your called... a professional
When you mix compassion and caring your called... a friend
When you mix bravery and selflessness your called... a hero
When you mix all three your called... A FIREFIGHTER!