Three more soldiers were injured in the "operation region", the defence ministry said in a statement after clashes with the militia on Thursday. It gave no further details.
Fighting broke out in several locations along the roughly 120-kilometre wide front where operations are focused, the Syrian Observatory for Human Rights and a Kurdish military official said Thursday.
Turkey on Wednesday began an air and ground offensive against the US-backed Kurdish People's Protection Units (YPG) militia.
Ankara has long opposed US support for the YPG in the fight against the militant Islamic State (IS) group, arguing the militia is a "terrorist" offshoot of Kurdish insurgents inside Turkey.
Turkey launches ground offensive against Kurdish forces in Syria
The ministry's statement comes after shelling from the militia into Turkey and Turkish military strikes in northern Syria, with casualties on both sides.
Seven Turkish civilians including a nine-month-old baby were killed in Turkey's border towns in Sanliurfa and Mardin provinces. Nearly 70 were injured.
Civilians flee
Thousands of civilians in Syria fled airstrikes and shelling as Kurds Kurds battled the Turkish military on Thursday, with fears mounting of a developing humanitarian crisis.
US President Donald Trump suggested that Washington could mediate between Turkey and Kurdish groups and a US official said Trump has asked American diplomats to broker a ceasefire.
French Foreign Minister Jean-Yves Le Drian called for an emergency meeting of the US-led coalition against the IS group.
Syrian Kurdish forces lost 11,000 personnel and played a major role in the years-long battle to eliminate the "caliphate" IS had set up in the region.
In scenes all too familiar since the start of Syria's war in March 2011, civilians were seen abandoning their homes on Thursday, in vehicles or on foot with their belongings on their backs.
The UN's Office for the Coordination of Humanitarian Affairs on Thursday estimated that 70,000 people have been newly displaced since Ankara launched its operation.
"We're heading to the countryside because we're scared of renewed bombing and intensified clashes," said Rizan Mohammad, 33, who fled the Kurdish-majority city of Qamishli with his family.
After launching the assault with airstrikes and intense artillery fire on Wednesday, the Turkish army and its Syrian proxies crossed the border into Kurdish-controlled areas.
On Thursday, Turkish jets carried out fresh strikes, the Syrian Observatory for Human Rights and a Kurdish military official said.
Fighting broke out in several locations along the roughly 120-kilometre wide front where operations are focused, they said.
Turkish forces and allied proxies captured 11 villages in the area, the Observatory said.
Fighting mostly centred around Tal Abyad - one of the main Kurdish-controlled towns in the area coveted by Ankara.
Clashes also raged around Ras al-Ain further west, the other main town in the zone that Turkish media reports say is the first goal of the offensive.
The Britain-based Observatory said at least 29 fighters and 10 civilians have been killed since the start of the assault.
AFP correspondents saw fighters crossing into Syria in dozens of vehicles.
Turkey, which staged two previous offensives into Syria since the start of the conflict, relies heavily on Syrian proxy fighters, many of whom once battled President Bashar al-Assad's troops.
President Erdogan wants a buffer zone to which he can send back some of the 3.6 million Syrian refugees his country hosts.
On Thursday he warned the European Union that the alternative was to allow the refugees to head to its shores instead.
"If you try to frame our operation there as an invasion... we will open the doors and send 3.6 million migrants to you," Erdogan said.
The SDF, which have little armour and no air force, are unlikely to hold out very long against Turkish firepower in the flat and open terrain they are defending.
"The question that remains is how far can Turkey advance before international or regional actors, stop it," said Nick Heras, an analyst at the Center for a New American Security.
The assault appeared almost inevitable after Trump announced on Sunday that US troops deployed in the area were pulling back from the border.
The withdrawal was implemented the next day, effectively clearing the way for Turkey's offensive.
Turkey threatens EU with refugee influx if criticises Syria military operation
One of the Kurds' last hopes is that the prospect of IS prisoners breaking out and regrouping with increasingly active sleeper cells will spur the world into action.
About 12,000 men linked to IS are held in seven detention centres across northeast Syria, according to the Kurds.
On Thursday, the Kurdish administration said a prison housing IS fighters in the region under its control was hit by Turkish bombardment.
COMMENTS (1)
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ