Forget the gags, too. They will usually teach a young horse to travel with his head and neck high and rigid, making it impossible for you to ask him to use his body to work in a particular gait. It is also very difficult to teach a horse lateral suppleness in a gag bit of any kind, which means that if you train him in one from the beginning he probably will turn like freight train. He won't back very well, either. What you may discover, however, is that the snaffle is not totally effective in teaching the horse to carry his head and neck where you want them in various gaits.
Snaffles tend to encourage a horse to travel with a low head and a stretched neck.
This will help if you have a naturally high headed, hollow horse that you are trying to teach to travel with a lower head and a rounder position to get rid of a pace or improve a running walk or fox trot. It will not help if you are trying to raise the head of a very low headed, trotty horse. Although you can raise a horse's head and tuck his nose in a snaffle by raising your hands and putting pressure on the reins, it is usually easier to switch to a bit that is designed to do those things than to use so much energy making do with a snaffle. A double bridle consists of a snaffle and a curb bit used simultaneously in the horse's mouth, with a set of reins to each. You can use the snaffle to lower the head and neck, or the curb to raise it and tuck the horse's nose.