At least 100 air strikes have hit the militias' positions.
President Recep Tayyip Erdogan ordered the military to begin Operation Claw Sword last Sunday, saying that air strikes would be followed by a ground invasion, pushing his forces further into Syria.
The operation followed a November 13 bombing in Istanbul that killed six, including two children, an attack Turkey blamed on the separatist Kurdistan Workers Party (PKK), which operates mainly in northern Iraq but also has positions in Syria.
Turkish air strikes have hit PKK targets in the mountains of northern Iraq, which Turkey considers a part of the same front in its war on Kurdish separatists.
Turkey’s operation — the third major offensive since 2016 — is mainly targeting the Syrian Democratic Forces, a US-backed, mostly Kurdish militia that controls large parts of eastern Syria.
“The forces that work symbolically with the international coalition in the fight against Daesh are now targets for the Turkish state and therefore operations have stopped,” said Mazloum Abdi, commander of the SDF.
Turkish soldiers are already present in districts of several provinces along the border with Turkey including Aleppo, Raqqa and Hasakah, occupying those areas with allied Islamist militias who are funded by Ankara.
Western countries have armed and trained the SDF in the fight against ISIS and the group was responsible — assisted by US air power and advisers on the ground — for expelling the terrorist group from Syria.
Major SDF victories include the battle of Kobani in 2015 and fighting the last battles against ISIS in Raqqa city and Baghouz in 2018.
ISIS prisoners warning
On Saturday, Mr Abdi told the BBC that the Turkish offensive could force his soldiers to abandon guarding open-air detention centres that hold nearly 60,000 women and children, most of whom are wives and children of ISIS fighters.
Separately, SDF forces are holding about 10,000 ISIS fighters in more secure facilities, although in January last year, ISIS launched a desperate attempt to free prisoners from Al Sina prison in Hasakah governorate, most of which is under Kurdish control.
ISIS used a suicide car bomb to breach the compound, where 3,000 prisoners were being held. About 140 SDF members and 400 militants were killed in the ensuing battle.
If the Kurds are forced to abandon these prisons, it "would lead to a second civil war in Syria and our counter-terrorism operations against ISIS would stop", Mr Abdi said. "As part of the international coalition, we fought and defeated ISIS, and what Turkey is doing will undermine all of it."
About 900 US troops are based in Syria, working in tandem with the SDF to co-ordinate against remnants of ISIS, which has been severely weakened since losing all of its territory, with its last bastion at Baghouz falling to US-backed Kurdish forces in early 2019.
On Wednesday, the US said a Turkish air strike had struck within 300 metres of their forces, and "directly threatened the safety of US personnel who are working in Syria with local partners to defeat ISIS and maintain custody of more than 10,000 ISIS detainees".
Turkey says the SDF is linked to the PKK, which has carried out terrorist attacks including suicide bombings within Turkey.
The US and EU have designated the PKK as a terrorist organisation and say their support for the SDF does not assist the PKK. Both groups said they had no role in the November 13 Istanbul attack.
The SDF said on Friday that as Turkish drones flew over the Al Hol camp that is home to tens of thousands of mostly wives, widows and children of ISIS fighters, some ISIS family members attacked security forces and managed to escape from the sprawling facility. The SDF did not say how many escaped but that they were later caught.
Kurdish authorities operate more than two dozen detention facilities scattered across north-eastern Syria holding about 10,000 ISIS fighters. Among the detained are 2,000 foreigners whose home countries have refused to repatriate them, including about 800 Europeans.