It sounds to me that your trainer is rushing you. If you feel unsafe at a canter, then you may need more time at a trot. Certainly, a gallop is probably not a good idea . . yet. But, if you need to hold the saddle, hold onto the fork, not the horn. You will be more stable that way . Think about pulling yourself DOWN, into the saddle, and BREATHE!!!! As you breath, you will relax, and let go of the fork, but keep your hand nearby, perhaps resting on top of your thigh.
You'll get there. Have patience, and confidence.
If you release any tension in the muscles of your body and remain balanced, it is nearly impossible for you to fall off. Given the "if", much of your weight will hang beneath the horse's spine on each side of the horse's body. You would need to lean pretty far to one side or the other in order to fall off. Releasing tension in the muscles of your body should also allow your body to naturally follow the motions of the horse while remaining balanced. Practicing such release of tension and following the motion of the horse at lesser speeds should help assure you that the same thing can be accomplished in the canter and gallop.