Rather than thinking of being brave which often seems to equate to recklessness, I would concentrate of developing confidence. Confidence in riding comes from a feeling of stability.
Stability is related to balance, and balance is related to the center or gravity. Balance in movement is also dependent on flexibility.
When a horse is cantering up hill, it is naturally better balanced with a lower center of gravity. That is why a rider generally feels more stable when cantering up hills.
Your falls and lost confidence have likely caused you become more tense and to cling more to your horse. Such tension in a rider usually creates tension in the horse which is often interpreted as testiness or misbehavior.
A more stable seat helps a rider relax, and relaxing creates a more stable seat. Just as a tense rider can cause tension in the horse, a relaxed rider can help his horse relax. Please understand that I am using the term relaxation as meaning lack of tension, not sloppiness. It is easiest to begin the relaxation process while sitting on the horse at a standstill.
As you sit on the horse, try to become aware of how you are sitting. Is your body balanced over your feet as though you were standing with your feet apart, legs slightly bent, and a horse just happening to be between them? If not, try to find this balance. Are you slouching as you sit? If so, don’t try to rigidly bring your body to an upright position. Instead, simply release any tension in the muscles of your torso. As you do so, these muscles should expand allowing the bones of your spine to stack one above another supporting the weight of your body while forming natural shock absorbing curves. You want your pelvis vertical rather than tilted backward or forward. You should be able to do this with little muscular effort. This should bring you into a position where you feel your weight broadly distributed over your seat bones and throughout your crotch – a very stable position.
Release any tension in the muscles of your crotch and throughout your legs and feet. You should feel your body sink more deeply into the saddle. Your legs should wrap gently around your horse’s body without muscular effort. While the stirrups stop the downward pull of gravity beneath the balls of your feet, your heels should be drawn lower. There is no need to push your heels down. Your inner thighs rather than the backs of your thighs should be against your horse which also helps stabilize your seat without muscular effort.
Once you have obtained this balance seat without tension, slightly squeeze and release your lower legs as you ask your horse to walk. To remain balanced when the horse is moving, the rider’s body must move to follow the horse’s movement. Having released the tension in your muscles, your body is free to follow your horse’s movement. Become aware of this feeling. Your seat bones move independently with your horse’s back flowing down, forward, up, down, forward, up. Your legs swing side to side with your horse’s barrel. Your lower spine moves every which way. Your relaxed upper arms pivot forward and back as your hands follow your horse’s head as it moves slightly to aid in balance.
Sense how stable this way of riding makes you feel. Sense how much more stable your horse feels as it releases tension in its muscles. Experiment to see how any minor changes in your movement can bring about changes in your horse’s movement. Try stopping all movement in your body and see if your horse stops moving. This may not work the first time you try it, but if you are patient and continue trying to relax and follow your horse, it will eventually happen.
So, what does all this have to do with cantering fast? As you learn to relax and follow your horse at the walk, your body learns to become more in tune with your horse’s body. This harmonious movement of the two bodies becomes natural. When you begin to trot and canter, you should find yourself better able to relax, balance, and move with your horse. Your horse should be more relaxed. Her movements should feel softer, smoother, and more flowing. You should both feel more stable with a lower center of gravity.