The Turkish army on Tuesday clashed with Kurdish militia in Syria in an operation that has already left three of its soldiers dead, as the United States voiced alarm that the offensive could endanger attempts to end the Syrian civil war.

Amid growing international concern over the four-day-old cross-border campaign into Turkey's neighbour, President Recep Tayyip Erdogan vowed that Ankara would emerge victorious.

Turkey on Saturday launched operation "Olive Branch" aimed at rooting out the Kurdish People's Protection Units (YPG) militia, which Ankara sees as a terror group, from its Afrin enclave in northern Syria.

The campaign has caused ripples of concern among Turkey's NATO allies, especially the United States which is still working closely with the YPG to defeat Islamic State (IS) jihadists in Syria.

In his strongest comments yet on the offensive, US Defense Secretary Jim Mattis urged Turkey to show "restraint" and warned it could harm the fight against the jihadists.

Mattis, on a visit to Indonesia, warned the offensive "disrupts what was a relatively stable area in Syria and distracts from the international effort to defeat" IS.

US President Donald Trump is expected to express America's unease in a call telephone with Erdogan on Wednesday.

French President Emmanuel Macron joined in the chorus of concern when he spoke by phone to the Turkish leader on Tuesday.

"While recognising Turkey's security needs, Macron "told his Turkish counterpart of his concern over the military intervention," the French presidency said.

- 'Until the last terrorist' -

Turkish artillery on Tuesday pounded targets of the YPG inside Syria, the state-run Anadolu news agency said.

Turkish drones were also carrying out attacks, state television said.

The Britain-based Syrian Observatory for Human Rights monitor said fighting was "very violent" to the northeast, northwest and southwest of Afrin.

As well as the artillery and air strikes, Turkish ground troops and Ankara-backed Syrian rebels have punched over the border several kilometres (miles) into Syrian territory, taking several villages, according to state media.

After intense exchanges, Turkey's forces took control of the hill of Barsaya, a key strategic point in the Afrin region.

The Observatory said 43 Ankara-backed rebels and 38 Kurdish fighters had been killed in the fighting so far. It has also said 28 civilians have been killed on the Syrian side but this is vehemently rejected by Turkey which says it is targeting militants only.

Sergeant Musa Ozalkan, 30, the first Turkish military fatality of the operation, was laid to rest with full honours in a ceremony in Ankara attended by the Turkish leadership.

"We will win and reach victory in this operation together with our people, together with Free Syrian Army," Erdogan assured mourners, referring to the Ankara-backed rebels.

Two more Turkish soldiers -- a first lieutenant and a sergeant -- were killed inside Syria on Tuesday in clashes with the YPG, the military said.

The campaign -- which Erdogan has made clear has no fixed timetable -- is fraught with risks for Turkey.

Two civilians have been killed inside Turkey in border towns in the last two days by rocket fire from Syria blamed on the YPG.

"This operation will continue until the last terrorist is eliminated," Turkish Prime Minister Binali Yildirim said.

Cavusoglu said in a television interview Turkey could extend the operation further to target other YPG-held areas in northern Syria including the town of Manbij and even areas east of the Euphrates River.

- 'Erdogan, Trump to talk' -

Ankara has expressed impatience with Western concern over the operation, arguing that the YPG is the Syrian offshoot of the Kurdistan Workers' Party (PKK) which has waged a bloody three-decade insurgency against the Turkish state.

The foreign ministry of Qatar -- Turkey's closest Gulf ally -- gave its unequivocal backing to the operation.

Critical is the opinion of Russia, which has a military presence in the area and a cordial relationship with the YPG but is also working with Turkey to bring an end to the seven-year-old Syrian civil war.

Erdogan said Monday that the offensive had been agreed with Russia but this has not been confirmed by Moscow. Cavusoglu denied there had been any "bargain or deal".

However many analysts argue that Turkey would never have gone ahead with the offensive without the Kremlin's blessing.

Erdogan and Putin late Tuesday discussed the operation by telephone, the Turkish presidency said, but the details of the call were not disclosed.

Turkey's previous incursion into Syria was the Euphrates Shield campaign of August 2016-March 2017, targeting both the YPG and IS in an area east of Afrin.

The Turkish security forces have meanwhile imposed a clampdown against anyone suspected of disseminating "terror propaganda" against the operation on social media.

Ninety-one people were detained in 13 provinces in Turkey, state media reported on Tuesday, after 24 people were detained in other cities on Monday.