First you need to assess your position when out on the trail. You shouldn't ride like an equitation class with your upper body straight. Riding cutters, you carry your weight in your feet, sounds ridiculous but once you have done it you can feel the difference. In order to do this make sure your stirrups are adjusted and turned properly to support your foot evenly. Then make sure the ball of your foot is supported.
The next thing you should do is sit more on your butt/ pants pockets. When you sit in good equitation position on your seat bones, you have excellent feel, but I think less stability. (Except for those amazing English riders who gallop over jumps. I feel more secure bareback than in an English saddle.)
Learn to let your upper body fold down a bit with the horse's movement. When they spook, you should bend a little closer to his neck rather than sit up and get stiff. This reduces the centrifugal force that is trying to throw you off while your horse spins away in a tight circle.
Push with your feet. If you try to wrap your legs around, all you're doing is playing catch-up to the horse who I guarantee is faster than you. So if the horse spooks left, your right foot should push down and out to send your body the same direction as the horse.
Now probably the most important, relax a little and always watch your horse's ears. Keeping your focus on the ears will help your body stay with him. If you look at the ground where you're afraid of falling, that is where you will go.
Ride holding the horn. There is no shame in that. It would be good to address why your horse is spooking, but at some will always get past a little startle if a deer or bird jumps out really close to them. It takes time and exposure. The best some horses can do in open terrain situations is that they choose to just stop, think, and listen to you rather than spinning for the hills or flying sideways.
The other thing you may want to consider is your saddle. Does it have the correct seat size for you? Do the seat and fenders have the right shape and hang for what you want to do? This can make a difference, so it is always good to try a few different styles if you can for comparison.
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