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What bit are you riding in now?
I would be careful to rule out any physical pain before proceeding. Once that is ruled out, consider any potential training gaps. There is no magic bit that will make a horse stop rearing or bucking.
That said, have you ridden her english before? What were you using then? What was ineffective about that option?
I typically start with the fattest single-jointed rubber loose ring to start and work from there. Buckers and rearers can be stopped by sending them forward with leg. If you have a bolter, you can easily pull up with one hand (horses cannot run very fast in a small circle). This is presuming you are a strong, confident enough rider to provide these corrections. Point being, a horse that misbehaves occasionally does not necessitate a stronger bit!
Most problems can be solved with effective hands and tons of flatwork. That said, I’ve found horses with some steering “rudeness” benefitted from a couple of rides in a full cheek with the same mouthpiece that they normally ride in.
I wish you the best of luck, it sounds like you guys are on the right path! My suggestions come from my experience riding a bunch of green beans, take it or leave it.
Get teeth done if you haven’t in last 6 months. Young horses lose caps, and need dental check ups every 6 months until adult teeth are all in.
You shouldn’t need a strong bit on a greenie. A full cheek or D snaffle with a bean type center is usually a good choice. Or a Mullen mouth snaffle.
Thanks for the reply. I will definitely try all of the things you have suggested. My goal is not to inflict pain. I just don’t want her to take off and get herself into worse trouble than what any bit could do. I have been looking at a couple full cheeks and think that is the way I will go.
If she takes off you could use a stronger bit but like others said, she should be trained to to take off instead of just being stopped when she does.
As for rearing, most horses that do it, do it because they want to move forward and the rider is stopping them so a stronger bit would likely make that worse.
A bit isn’t going to stop your horse if you haven’t taught your horse the meaning of “whoa!” from the ground first. Horses aren’t born with brakes, they have to be installed by their riders (trainers). If you don’t know how teach your horse the cues for stopping then you should send her out to a professional who can.
You also need more training to learn how to stop a runaway horse! I know a lot of people think that an emergency dismount is the answer, but it isn’t; it puts both the horse and rider in even more danger of bodily harm. There are several ways to stop a runaway horse and every rider should be taught by their trainer/riding instructor how to handle the situation before it occurs.
Poorly trained horses can and will learn how to evade even strong bits and blow through them. Your best insurance against something bad happening is knowledge…
Getting on a horse that doesn’t know how to stop is like riding in a car that has no brakes! Not a smart thing to do… :no:
You sound like you are recently new to riding and your very very young mare is also new to being ridden and also very immature. Horses very rarely do things randomly and I think the problem is that she isn’t broke or isn’t very broke.
I think that you should put her in training with a professional while you take lessons on both her and more experienced horses. Her issue is a lack of training and not bitting.
Agree with above. That said a full cheek snaffle with a plain mouthpiece might be a good place to start
Agree with Lace - Full cheek snaffle for steering. Stronger bit only if absolutely and unequivocally necessary.
Love the description of steering “rudeness” as well 
If all else fails try a few training rides with someone whom is familiar with babies.
Never “bit up” a green baby, too much risk of ruining them and it will give the opposite effect because they don’t yunderstand fully the bit cues(so you may end up with more reactions out of confusion).
I prefer D french link, ride in an enclosed arena.
