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Always train the horse with a mild snaffle bit before moving on to a more severe option. Stronger bits can provide more control, which may be useful for horse shows. Some excitable horses will resist stopping with a mild bit, but focus on ground work training first. Some spirited or sensitive horses will object to a harsh bit, and continue misbehaving.  Think about your typical riding activities. You'll typically want a mild bit for riding in the school, and perhaps a stronger bit for more strenuous activity. Check the bit regulations of any horse competitions you plan to enter. Beginning riders are more likely to harm the horse with a strong bit. You should have a fair amount of riding experience before you try to use one.  Even then, ask your trainer for advice on choosing a bit, and on adjusting your rein use to adapt to a new bit. There are many variations of bits, and you may need to try several before you find one that works. Most English riders just move to a snaffle with a harsher mouthpiece or ring. Western riders may try a different style of bit altogether, such as the curb bit. These use leverage to increase the amount of pressure from your pull. The Pelham and the Slotted Kimberwick are two common options which use a curb chain to provide more control over the head. Bitless riding is becoming more popular, although it is not yet legal in most competitions. Even if you use a harsh bit some of the time, consider training without a bit occasionally. This will improve your own balance and strength, and teach you to direct the horse with cues from the "seat." This may even solve some behavioral issues, as many horses calm down when the mouth is no longer agitated. Bitless riding is allowed in most jumpers, polo, and endurance competitions. For other disciplines, you will need to compete with a bit — but you may still benefit from the training.
Decide whether a stronger bit is appropriate. Consider your own riding ability. Choose the bit. Train the horse without a bit.