AMUDA, Syria—Amid the chaos of Syria’s war, the Kurds have carved out a semiautonomous region called Rojava that is home to about four million people, is as big as Belgium and stretches nearly the full length of the 565-mile border between Syria and Turkey.

The emergence of Rojava also has added complexity to a region in turmoil, bringing resistance from outside and dissent from within.

Rojava’s...