The full-cheeked snaffle is available with a variety of mouthpieces, such as the french link and the slow twist.
The cheek pieces can be helpful in refining the turning aids and for preventing the bit from being pulled through the mouth.
I use a french linked full cheek snaffle with keepers (small leather loops), which fix the bit in a position which keeps the link in the center of the mouthpiece flat on my horse's tongue. When used without the keepers, it will twist in his mouth and give the bit the same action as a Dr. Bristol mouthpiece, which is more severe than the french link.
The keepers, by fixing the bit in alignment with the cheekpieces of the bridle, also add a small degree of pressure on the poll when the rein is used.