coloredcowhorse has given you the textbook description of the progression.
Since you are a dressager, I’ll point out two things.
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The bridle horse world has as much philosophy in it as the dressage world. There are plenty of old skool books that will help you. Being in California, you are also in one of the epicenters of the bridle horse tradition. See if you can find a mentor.
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The western horse has a very different relationship to the bit than does a dressage horse (and I’m thinking more of the German rather than French approach). What it means for a western horse to be “in the bridle” at any point in his career does not mean pounds of pressure in your hand.
Even with the snaffle horse/baby, you are not asking him to physically push into the bit, “to seek the bit.” That’s why these snaffles are loose rings (the bit is unstable-- there’s nothing to rest on). That’s why people starting colts tend to hang the bit low in the horse’s mouth. They want him to “pick up” and “carry” the bit. They also want him to be able to get away from it. That’s why some western people move into pretty thin bits ASAP during the snaffle phase. Maybe these guys just didn’t begin the transition to a mecate soon enough. I get the impression that that middle phase of the bridle horse’s training doesn’t get a lot of airplay by modern trainers. Or the people who are quick to go from snaffle to a Tom Thumb (not a fan) to a leverage bit didn’t actually teach the horse what they wanted him to do with his body during the snaffle phase quite fast enough or weren’t effective.
In any case, the goal for the western horse is to have the bit be neutral in his mouth at all times. It should be decoration, the very last aid used in a progression of them. What they do want is a horse that goes off your seat, weight and leg. The snaffle is just a means to an end. IMO, western horses get a whole lot more broke to those aids sooner in their training and it’s a beautiful thing. At bottom, western trainers want to reverse that ratio of weight carried on the front- and hind end just like dressagers. They spend years getting that, just like dressagers. But they go about it in a slightly different way. On the surface, it looks like they “cheat” and get that done with some scary hardware and a kind of riding that looks too quick and too advanced.
IMO, by the time the horse is going in a mecate (understood as a transition to a leverage bit) he already understands that he can never drop his shoulder. Sure, he’s still building the strength to do that. But the leverage bit is a “signal bit”-- it can’t force a horse to lift his shoulder and squat on his hind end. (Well, it can temporarily-- and that’s the problem.) In theory, the leverage bit is a “reminder” about what you already taught the horse about carrying himself and listening to your body. So the people doing a lot of crude training with a curb bit have cut corners. IMO, and I’m an outsider, I think big training problems get fixed with a snaffle, not a leverage bit.
People have spade bits custom made, spend a lot of time thinking about weighting reins or choosing the right thickness/weight of their split reins because all of this combined with the horse’s conformation determine how the bit will sit in his mouth, and how any movement of the rider’s hand will create a particular signal. “Conformation” here involves everything from the horse’s tongue and bars to shoulder and withers. I’ll bet the horse’s personality and tendency to go forward or get behind your leg gets figured into it, too. Old skoolers used to have just one bit fabricated for their bridle horse and he would wear it for the rest of his life.
So I ask all dressagers-- who wouldn’t want a horse trained like a bridle horse? They are what anyone would want in an expensive form of transportation, IMO. They are physically easy to ride. On the mental side, they have the sense that they must answer any aid, pronto.
Have fun with figuring out bridle horses in California. You have a great opportunity!