KAYSERI, Turkey — As the muezzin heralded the noon prayers on a recent Friday, a small army of workers fanned out from an industrial park to take their places on mats in a nearby mosque. Fifteen minutes later, the prayers were over and the teachings of the Koran gave way to the demands of the factory floor. “In European countries, workers take a 15-minute smoking break; here we take a 15-minute prayer break,” said Ahmet Herdem, the mayor of Hacilar, a town of 20,000 people in central Anatolia, a deeply religious and socially conservative region which has produced some of...