ISTANBUL—By almost any standard, Syrian refugee Abdurrahman is doing better than most of those who fled the war. The 68-year-old from Latakia recently opened a spice shop in Istanbul, his third small business in the city, and he now speaks enough Turkish to look after his customers.

But Turkish hospitality for refugees like Abdurrahman is limited. He isn't allowed to buy any property in Turkey. He would face high hurdles if he wanted to become a Turkish citizen. And he worries that the Turkish government isn’t doing enough...