You would probably get more answers in the training section.
You need to ride with your seat more. From what you have explained, this is not a lesson horse that you pull on the stop. I think the more horses there are, the more you worry about stopping your horse and when you start thinking about it, you stop riding with your seat and you start pulling on the reins. If I pulled on my horse's mouth with both reins to slow him down, he would only go faster and through his head in the air giving me no control-most horses will do the same. If you want to do a tug of war battle with a one thousand pound animal, who will win, you or your horse? DON'T PULL!!! When you pull it locks a joint in their back, their neck comes up, they brace with their bottom neck muscles, they hollow their back and then guess what?-you have no control.
You should never take leg aids off a horse. Leg aids are used to slow down, back up, and move laterally. When you ride all you aids must be harmonized and if you take you legs out of the equation you have no base to use your core off of and then you only the your hands. Pulling a horse's mouth will get you nowhere.
This is what you must do to slow him down if he starts going fast. I'll start with how you should be in the first place which should prevent it and if he does go fast I'll tell you what to do. However, it would be 1000000 times better if I could stand in the ring with you and watch you. Is it possible that your instructor could give you 10 minutes of the lesson to help you on that? One on one?
1)Ride balanced over your leg(ear, shoulder, hip,heel=straight line). Support yourself with your core, do not hollow your back, have a straight back. Pull your elbows closer to you hips. Look up and ahead, not at the ground.
2)At the trot: Use leg to push the horse into the bit, but not on the forehand. Use your legs and upper body to keep the horse off the forehand. Sit up and look up, that keeps the horse up. If you look down the horse goes down on its forehand.
3)Go onto a 20m circle. Keep steady contact with the outside rein and shape the circle with the outside leg and rein. Do not pull on the inside rein. Sit balanced with your shoulders straight and facing the direction you are going. Shape the horse with you thigh and knee and keep him on the circle with you calf and heel.
4) to control his pace move your body more freely or less freely, "close" the aids or "open" the aids. You don't want him in a short strided tight frame but a loose, relaxed frame. If you ask him to canter from a short strided tight frame, he will go much faster in the canter instead of "rolling" his shoulders in the canter. By "opening" the aids, I mean open up your shoulders, loosen your thighs, give with your inside rein, and with your seat, calves, and heels "push" the horse forward. By "closing" the aids: hold the outside rein firmly, close your thighs, and think of stopping with all your aids at once, but do not loose your leg! When you are in the saddle as you post, close your aids=half halt. Then release and do it again. To get a good canter he must have free moving relaxed and forward trot.
5)Cantering: Sit in the saddle, you must
stay balanced over your leg. Move you outside leg back and hold it against the horse, ask him to move up
into your hands. Sit upright with you core tight and holding the horse off of his forehand. When a horse is on its forehand, it is unbalanced and moves faster. Continue to hold the outside rein.
6) If he speed up in the canter, half halt: sit deeper, hold the outside rein and slow him with all your aids and most importantly your core. Your legs support your core and is part of all your aids, it also holds the horse together, and keeps it balance. You need the horse forward and off its forehand even if you are trying to slow it down. It must be balanced to slow it down, your legs help shape him and balance him. You must half halt, release, half halt, release. When you release, you are not letting go of all your aids, you are just stopping the closing of your aids. When he slows down. When you get him into a frame, you don't want to loose contact. Your contact should never be pulling, only connecting the rein aids with aids from your legs and body. To get him into a frame, you should still have your outside rein on(not pulling, only connecting the bit to your arm), and the outside rein on because your are still shaping him to to 20m circle. Squeeze your fingers of your inside rein and pull slightly, then release. All you want is to have him turning his nose slightly when you do that. When he is doing that, he is getting supple at the jaw. When you release, lower your hands and then repeat. To help turn his nose and supple his jaw you should use some inside leg to help with bend and supple his poll but your outside leg and rein continue to shape the circle. He is always moving forward and slowing him down does not
mean bringing him onto his forehand!
A tip for slowing him down at the canter is when to do the half halt. Do the half halt when his pace has pushed you into the saddle, that is also when his legs are all on the ground. When you lengthen the time his legs are on the ground you are slowing him.
This isn't just about slowing him down, it's about balancing him and connecting your aids.
Please tell me if you have any questions!