They wear whatever bridle color they want for racing. Many people use white because it can go with any color silks. Some buy a bridle that matches the main color in their silks. I use white because my colors are hot pink and white but I think the white shows up better. They do make white leather as shown in the link above but I much prefer nylon. I think it looks better and is safer.
I'll always take leather over nylon--I don't like things that won't break or cut easy if they have to. I use a racing bridle (in dark brown leather) and it's reinforced enough I don't worry about it wearing or snapping, but it's still leather and if it gets snagged it will break.
Not a fan of white anything--I like how it looks, HATE cleaning it! Never had any luck cleaning nylon halters (when we used them.) They came out looking dingy.