Kudos to you faster horses, took all the words right out of my mouth!
The reason that your horse is not turning at the gallop because he is stretched out, weight on the forehand, not collected, not listening to you and you are not in control. Could make for a very embarrassing barrel run! Bottom line is that you are not safe.
I would go back to the snaffle, or even the halter and stand beside his shoulder facing front. Put a very light pressure on the rein on that side and wait. When he brings his face even the slightest around to you, release. You have to concentrate in order to release immediately. Repeat until he gives his face all the way around to you. Then do the other side. Even though this is a standing still exercise, its hard for the horse whose neck muscles are not used to it.
Then stand facing his shoulder and add rein pressure on that side up by his withers and look at his shoulder. When he gives, hold the pressure and don't release until he moves his front foot away from you. What is happening is that the horse wants to relieve his neck muscles and he learned that when he gave his face to you in the first lesson, he was relieved by the release (rewarded) In the 2nd lesson, the horse is looking for release, but when you don't release, he'll move his feet to find release...then release.. When he has one side down pat, do the other. When you progress to getting him to turning a small circle and coming to face you, then you release. This is the one rein stop and if trained well, it will give you the necessary brakes that you need in all situations.
You are teaching him to not only listen to you and the rein, but to give to it, train his muscles to bend and be more flexible and you are also training him to become ambidextrous and able to give equally on both sides. You should never apply rein pressure on both reins at the same time.
You won't be able to do all this in one lesson, but I still think you will be surprised at how fast it can go. Tell him when he's good, that really helps speed up the learning process...encouragement.
When you arrive at this point, get on him and do the same thing again. If you have done your homework, with the slightest rein pressure he should swing his head right around to your foot, either side. Forget the curb and definitely use the snaffle. Then go out and find some bushes to go around out on the trail as this will help the horse make sense of why he learned all this. Do nice circles around those bushes and don't let him drop his shoulder. Good luck!
