I would still ditch the Tom Thumb regardless, but as 1RedHorse already said, this really isn't a bit issue.
Especially in your first "run" when you are trotting around the first barrel, watch how his butt swings around his front end the whole time. He should be driving from his hind end, not swinging it around. This could be helped, in part, by adjusting your reins and hand position. Your reins are too long and you are holding your arm straight out to the side. Your hand should be more toward your saddle horn area; not straight out in the air.
While this particular rider is leaning forward a bit in her turn, check out her hand position. It is quiet, it is low, and it is only assisting the horse -- not pulling to the outside. Granted, every horse will run a little different and needs different "help", but in general, you want quiet, low hands.
You'll need to work on alot of drills to get your horse to use his hind end, and not swing it around. I'd go back to some slow work and slow drills AWAY from the pattern.
Dena Kirkpatrick's method talks a lot about "perfect circles" which is getting the horse to stay in a rounded frame, engaged hind quarters, on one inside rein in a circle. She has it on her DVD, but she does have a lot of free videos on YouTube. Here's one of them that can get you to the rest. I would suggest you watch them all.
You also need to finish
all of your barrel turns. Especially on the 1st run-through, second barrel, you drift out instead of finishing the turn. When you get done with your barrel pattern, if you had a fresh clean arena, your tracks should mirror this pattern, with straight lines from barrel to barrel. Not drifts or arcs.
I won't comment at all on the last run on your friend's horse. Maybe practice a bit on some of these same things with that horse, next time, before making a run?
And not meant as a snotty remark, but I'm sure if you can afford a $50 to $100 entry fee at a rodeo/race, not to mention the gas/diesel to travel there, can't you afford to buy a proper bit for your horse? And skip the next race. Or at least try to borrow a bit to try from a friend. Tom Thumbs are best used as coat hangers.
As far as what bit to try, I'd go back to a regular ol' O-ring snaffle for now. Your horse has some issues with using his body correctly and going back to the basics is going to be your best bet.