DIYARBAKIR, Turkey—Just three months ago, busloads of European tourists thronged this city’s walled old quarter, admiring its ancient churches and mosques that had just been declared a Unesco World Heritage site.

These days, the entrances into the warren of Old City streets are blocked by ditches to stop police armored vehicles. Sheets hung from wall to wall are meant to protect against snipers. Barricades made of cobblestones and sandbags are manned at night by young Kurdish fighters armed with Kalashnikov rifles. The local...