This sounds like your horse just needs plain ol' training. I wouldn't even have him on the barrel pattern itself right now, as there are plenty of things you can do to train him on basic riding fundamentals, that can be carried over to barrel training.
You should be able to stop him easily from any gait, without fighting his mouth. He should back up readily and relaxed, and giving his mouth properly to the bit. You should be able to walk, trot, and canter on a loose rein while staying in perfect control of him. He should be able to respond to your leg aids and do sidepassing, leg yields, and diagonals. He should respond to your seat (that you are using to slow him down and/or stop). He should both direct rein and neck rein easily. He should know both simple lead changes and flying lead changes, as you need flying lead changes for barrels (and he should be attuned to your legs anyway). You should be able to do rollbacks with him, as well as turns on the haunches and turns on the fore.
In a nutshell --> He should be 100% broke to ride and you should be able to control any part of his body at any time at any pace.
So the fact that he won't turn when you ask him (barrels or not) means you need to go back to training basics and fix that training issue.
An O-ring snaffle is fine. Actually, that's what I prefer to start my horses in. You only bump up to a shanked bit when they are ready to be fine tuned and lighter in their cues.
Without actually seeing a video of what's going on, the best I can assume is that YOU are doing something wrong with rider error. Are you cueing and releasing at the appropriate times? Let's say we are going to do a turn to the left. You'll keep your left leg off of him (to "open the door"), you'll give pressure with your right leg, give direct rein pressure on the left, and give a supporting neck rein with the right rein. Don't pull harder .... just stay consistent. The very instant he responds correctly by turning to the left, even the smallest bit, you need to immediately remove all your cues. That is his reward for doing the correct thing -- you remove the pressure.
If you are not cueing him right, or if you are not releasing the pressure immediately, all you are doing is confusing the heck of out him, and he just learns to blow through your cue or ignore it alltogether.
As I said, I don't know that your riding is the problem, but more often than not, the rider is at fault. Hard to say what exactly is going on without seeing a video.
And if he gets bored easily, don't sit and do the barrel pattern over and over again. Just go trail riding and randomly work on things. If you see a tree stump, go turn around it like it is a barrel. Walk toward a fence and do a rollback when you get to it. Do serpentines up and down the ditch. As him to sidepass over a log. Back a ways down the trail instead.
In the arena, you can set more than 3 barrels up in random places around the arena, and just randomly go from one to the other turning them. Turn some of them multiple times. Go past some and don't turn them. Etc. Just keep it different.
You have lots of time to pattern him on the pattern. Sounds like you need to work on basic riding fundamentals first.
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It is not enough to know how to ride; one must know how to fall.