In Diyarbakir, the largest city in the region, more than 5,000 demonstrators gathered, with some throwing stones at police and several dozen being detained.
In Yuksekova, a small town near the borders with Iraq and Iran, police used tear gas and water cannons to disperse a crowd of several thousand that had set up roadblocks, the Anatolia news agency reported.
A car bomb which went off during the protest injured a 28-year-old man, Anatolia reported, adding that police had launched an investigation into what type of explosive had been used.
Quashing protest
It said police had made "numerous" arrests and that shopes had stayed closed.
Police had also clashed with Kurdish demonstrators on Friday in Istanbul and other cities.
In the Umraniye district of Istanbul water cannons were used to disperse the crowds, while in nearby Kucukcekmece petrol bombs damaged a shopping centre, Anatolia reported.
The pro-Kurd news agency Firat said police had put down protests in the cities of Mersin, in the south, Sanliurfa in the southeast, where vehicles were set on fire, and Van and Varto in the east.
Mistreatment allegation
Ocalan's lawyers had reported he had been assaulted by a guard and threatened with death in his island prison of Imrali where he has been held in solitary confinement since 1999.
Similar reports in the past have stirred anger among Kurds who look on Ocalan, head of the separatist Kurdistan Workers' Party, as a hero.
Arrested in Kenya in February 1999, he was sentenced to death by a Turkish court but the sentence was commuted to life imprisonment in 2002 after Turkey abolished the death penalty.
The PKK is listed as a "terrorist" organisation by the EU, the US and Turkey.
About 44,000 people have died since its conflict with the Turkish state began in 1984.