They are fighting for control of areas near the border city of Jarablus, recently won back from the self-proclaimed Islamic State, IS.
Turkey says 25 Kurdish militants were killed in the air strikes, while a monitoring group claims at least 35 civilians died.
In Syria's northern countryside, Turkey steps up its cross-border military campaign against Kurdish fighters.
Operation Euphrates Shield, now in its sixth day, is designed to stop the advance of IS fighters and Kurdish militia.
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On the first day of the campaign, Turkey and its allies won back the border city of Jarablus from IS.
Now, with a major focus on Kurdish-controlled villages around the town, Turkish warplanes have roared across the border at daybreak.
They struck at what security sources say were sites owned by the Kurdish People's Protection Units, or the YPG.
Turkish president Recep Tayyip Erdogan says his country's pursuit of militant groups will be unrelenting.
He was speaking at a rally of thousands of flag-waving supporters in the city of Gaziantep, where a suicide bomber killed 54 people at a wedding this month.
"We will give every effort to support efforts to cleanse IS from Syria and Iraq. That's why we are present in Jarablus, and that's why we are present in Bashiqa. We will not avoid taking responsibilities in other regions if necessary. We have the same resolve, with the separatist PYD's Syria wing. Our struggle will continue until we root out this terrorist organisation."
President Erdogan wants to stop the YPG expanding its territory inside Syria, amid fears of a Kurdish state being created.
But the battle has strategic difficulties.
While Turkey and the Kurdish rebels are bitter enemies, they are both also allied with the United States in the fight against IS.
The Turkish military says it only killed Kurdish fighters that it labels terrorists in the latest air strikes.
But the British-based Syrian Observatory for Human Rights says dozens of civilians have died.
It claims fighting in two villages south of Jarablus killed at least 35 people, with scores more wounded.
They would be the first significant civilian casualties since Turkish tanks and troops began the offensive.
Four local fighters were also reportedly killed.
President Erdogan says the cross-border offensive will give those who fled the city of Jarablus the chance to return home.
"Jarablus has been emptied of Islamic State militants. The people who are the owners of Jarablus are starting to settle there now. We will provide all kinds of support and assistance to those who are in Gaziantep and want to return. Now, everybody should know his or her place. We have given every kind of support to the Free Syrian Army, and they have undertaken the necessary operation."
Turkish forces have also moved west towards further IS areas.
