Fear is more an emotion than a rational thought. I don't gain much from telling myself to not be afraid. I find it more productive to actively seek ways to reduce the risk. When I know
I'm safer, I feel safer.
Bitten or stepped on? Hasn't happened to me yet, but I own my own horses so I get to choose how they are treated.
Falling? I can't afford it. I spend too much time riding on pavement or surrounded by cactus. It isn't 100% preventable, of course, and I could fall the next time I ride. But...
After my first (and to date, only) fall, I switched from an English saddle to an Australian one. An Australian saddle with an American horn added is one tough saddle to fall out of! Was I cheating? Yeah. So what. It worked. Later switched to a western saddle, and more recently to an Abetta. The Abetta saddle is much grippier, and thus much easier to stay in, when
my horse hits the fan.
If your horse is unpredictable, then I think being a little "behind the horse" helps with staying on. Others disagree, which is OK. Do whatever you believe (or a good instructor tells you) will help you stay on an unpredictable horse.
"Speed and surprises don't mix well with me
" - so practice them. Canter in a round pen if needed, or on a lunge line. Practice lots of transitions - changes in speed or direction. Practice turns. Do it while standing in the stirrups, which will reveal any flaws in your balance. If need be, do the latter in a western saddle so you can keep one hand on the horn to help you balance until you don't need any help balancing.
Lastly, it is OK to get scared and dismount before
the horse gets scared. It is an emotion. It doesn't have to be rational. But for me...riding while scared only built up memories of being scared on horses, which increased my fear. Once I felt free to dismount, and then mount up 10 minutes later when the fear passed, I started overwriting memories of being afraid on horses with memories of having good times on horses. Pushing thru my fear didn't help me any more than it helped my spooky horse. BOTH of us needed to build memories of good times and staying safe.