It's not a matter of middle east versus the west, it's a matter of desert versus the plains. Deserts are generally windy places, where a wide brimmed hat would keep tugging your head or be snatched right off by the wind. Better to wear a turban that can be wrapped around the head and tied in place. It offers smaller surfaces to catch the wind.

It's not restricted to the middle east. You'll find similar turbans in northwest India or Pakistan (Thar desert, which generates a hot, dry wind that blows all summer long, notorious in those parts). Contrast with wide-brimmed straw hats popular in Vietnam or parts of southeast Asia, which are equally sunny but far less windy.