When we’re talking about horses, harnesses, restraints, or horse trails (i.e., bridle trails), the word is bridle. Its origins are in the Old English word bregdan, meaning to braid.
When we’re talking about a woman who is getting married (i.e., a bride), the word is bridal. This word comes from the Middle English noun bridale, meaning wedding.
Examples
Some clues that horses might be having dental problems include loss of feed from the mouth, lugging on the bridle, head tilting or tossing, resisting the bridle. [The Horse]
The Gazette and Herald newspaper is reporting that he also raised concerns about the bridle that was being used on this particular horse. [Ashley Ainsworth]
Some vendors offered traditional services, such as bridal wear, wedding photography, DJ services and reception catering. [Keokuk Gate City Daily (link now dead)]
Bridal gowns are going 3-D, although we won’t need to wear clunky glasses to appreciate them. [Plain Dealer]