Turkish police have launched criminal investigations into Kurdish lawmakers and detained scores of people, accusing them of criticising the military's incursion into Syria on social media, state media reported.

Police also fired water cannon and detained dozens of activists in the mainly Kurdish southeastern city of Diyarbakir on Thursday at a protest against the cross-border assault.

Turkey pounded the Kurdish YPG militia in northeast Syria for a second day on Thursday, forcing tens of thousands of people to flee and killing dozens. Ankara deems the YPG a terrorist group over its links to militants who have waged an insurgency in Turkey.

Police in riot gear fired water cannon and then detained protesters at a rally in front of the pro-Kurdish People's Democratic Party (HDP) headquarters in Diyarbakir, the biggest city in the southeast. A HDP party official said police detained 25 members there, including its Diyarbakir provincial leader.

Earlier on Thursday, authorities began investigating HDP leaders and detained 21 people for criticising the military offensive online, state-owned Anadolu news agency said.

While most of Turkey's opposition parties have backed the operation, the HDP has called for the offensive to stop, describing it as an "invasion attempt".

HDP co-leader Sezai Temelli said on Wednesday the operation was an attempt by the government to drum up support amid declining public backing. Prosecutors launched an investigation against him and HDP's other co-leader, Pervin Buldan, over their remarks about the offensive, Anadolu said.

It said Temelli and Buldan were accused of "carrying out propaganda for a terrorist organisation" and "openly insulting Turkey's government". Three other HDP lawmakers were being investigated over similar charges, the news agency said.

The government accuses the HDP of ties to the Kurdistan Workers Party (PKK) militant group and thousands of its members have been prosecuted for the same reason, including its leaders. The HDP denies such links.

Hours after the operation started on Wednesday, authorities also launched investigations against 78 individuals who criticised the offensive on social media, Anadolu said.

On Thursday, 21 people were detained in the southern province of Mardin for their social media posts. The suspects are accused of "provoking the public to hatred and animosity" and "carrying out propaganda for a terrorist organisation", Anadolu said.

Late on Thursday, police detained another 11 people linked to the HDP who were protesting in central Ankara against the incursion, Anadolu reported.

Authorities launched similar investigations after each of Turkey's two previous cross-border operations into Syria. More than 300 people were detained for social media posts criticising Turkey's offensive into northern Syria in January 2018.