ISTANBUL—Turkey’s Kurds defied the state of emergency to stage nationwide protests on Wednesday against the detention of a leading Kurdish politician, as the government’s post-coup crackdown increasingly ensnares the restive minority and fuels an insurgency destabilizing the North Atlantic Treaty Organization member.

Hundreds of people gathered in Diyarbakir to protest the Tuesday night arrest of Gultan Kisanak, who was elected mayor of the symbolically important Kurdish city with 55% of the votes in 2014, and her co-mayor,...