Bridle

A bridle is a piece of equipment used to direct a horse. As defined in the Oxford English Dictionary, the "bridle" includes both the headstall that holds a bit that goes in the mouth of a horse, and the reins that are attached to the bit.

Headgear without a bit that uses a noseband to control a horse is called a hackamore, or, in some areas, a bitless bridle. There are many different designs with many different name variations, but all use a noseband that is designed to exert pressure on sensitive areas of the animal's face to provide direction and control.

https://en.wikipedia.org/wiki/Bridle