The answer is: Sometimes.
The "war bridle" as it is called was certainly used by some Native Americans.The strap was made of rope, horsehair or leather, and it was used only on well trained horses who needed very little rein action - you cannot steer a horse by direct reining in an Indian bridle.
(This is not the same thing as a modern war bridle, which is run over the poll).
The Indian war bridle was actually not the mild device people romanticize it into.
However, the Native Americans used a variety of riding styles and equipment. On the Plains they copied Spanish style saddles (and, like the Spanish, mounted from the right).
Indian war bridles are still sometimes seen occasionally, but you're more likely to see traditional western style bridles made in styles and with decorations that fit the fashions of that particular group.