(CNN) Turkey's military incursion into northern Syria against Kurdish militia opens yet another front in the seven-year Syrian conflict, and risks giving ISIS breathing room just as it was being suffocated.

Turkey has long warned that it will not tolerate control of much of its border with Syria by the "terrorist" Kurdish YPG militia -- even as the US has bolstered its support for the YPG as its proxy in the fight against ISIS.

In effect, one NATO member is trying to take down a group which is trained and armed by another and which has done much of the fighting against ISIS -- while Syrian President Bashar al-Assad fumes on the sidelines.

To Turkish President Recep Tayyip Erdogan, the YPG are indistinguishable from the Kurdish separatist group in Turkey, the Kurdistan Workers' Party or PKK.

The offensive -- euphemistically called "Operation Olive Branch" -- began with dozens of airstrikes Saturday. On Sunday, Turkish troops crossed into Syria, supported by rebel factions. The commander of one rebel group supporting the Turkish offensive said 13,000 fighters were involved.

Turkey's operation targets the Kurdish enclave of Afrin, adjacent to the province of Idlib. Turkish forces are deployed in Idlib to police one of the four '"deconfliction zones" designed by Russia to reduce the fighting in Syria. But the Turkish military has opted to use its presence in Idlib to open a new front against the Kurds.

The Turkish military said airstrikes had destroyed 45 targets, including barracks and weapon depots. The YPG says airstrikes have targeted at least 100 locations. But it also claims that resistance has blunted the early stages of the ground offensive.

Turkey's Deputy Prime Minister Bekir Bozdag says the Turkish border town of Reyhanli was hit by a YPG missile fired from inside Syria, killing one and injuring more than 30. The Turkish border city of Kilis was also hit by rockets fired from Syria on Sunday, according to Turkey's state-run news agency, Anadolu.

In statement released Sunday, the Syrian Democratic Forces (SDF), denied it was to blame for the attack on Reyhanli. "Allegations that we have launched attacks across the border are false and a pretext for Turkey to bring its military forces and its extremist opposition groups onto Syrian soil," the statement said.

Advance warning

Turkey notified both the US and Russia of its intentions . Turkish Chief of the General Staff Hulusi Akar was in Moscow last week to ensure there would be no unintended consequences; Russia has powerful anti-air radar in northern Syria. Some analysts believe Turkey has struck a bargain with the Russians -- help "deconfliction" in Idlib in return for a free hand in Afrin. Russia withdrew its modest military presence in the area ahead of the Turkish operation.

For its part, the YPG says it has been betrayed by the Russians, with whom it has previously cooperated. "We also hold Russia responsible for these attacks, and we hold Russia responsible for any massacre of civilians," it said in a statement Sunday.

Now everything depends on how extensive the Turkish operation is. If the goal is a limited buffer zone along the border, conflict may be contained. But if Turkey wants to seize the city of Afrin -- and then launches a second front to seize the town of Manbij further east, another war within the war in Syria will erupt.

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Mattis: 'We'll work this out'

The US is calling for restraint by all sides but has not (unlike Iran) demanded an end to Turkish military operations. The State Department said Sunday that the US was concerned about "the plight of innocent civilians" and called on Turkey to "ensure that its military operations remain limited in scope."

US Defense Secretary Jim Mattis, speaking on his way to Asia, said: "We are working now on the way ahead. We'll work this out." Mattis added that "Turkey has legitimate security concerns" in northern Syria.

The current evidence suggests Turkey's aims are far from limited. Erdogan has demanded the surrender of Manbij, where US forces are present. The Turkish President insisted Saturday: "Beginning from the west, step by step, we will annihilate the terror corridor up to the Iraqi border."

The US has justified its support for the SDF, an umbrella group in which the YPG is the largest player, as a critical part of the strategy to prevent ISIS re-emerging. The dispute has driven relations between Washington and Ankara almost to breaking point. For Ankara, the last straw was a US plan announced earlier this month to train 30,000 SDF troops to patrol the Turkish and Iraqi borders.

Erdogan tweeted: "The US has now acknowledged that it has established a terror army along our borders."

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Provoking a wider war?

The Washington-based Institute for the Study of War noted at the weekend that "Turkey's operations threaten to provoke a widening Turkish-Kurdish war that could unravel the US stabilization effort in eastern Syria and force the US to reconsider support for the YPG."

For Erdogan, sending in the troops plays well to his nationalist base in Turkey and has won the support of opposition parties. The leader of the right-wing Nationalist Movement, Devlet Bahceli, declared Sunday that "the operational partner of the United States should be rooted out," and that the offensive would counter "the [Western] attempt to reshape the Middle East."

The leader of the main opposition Republican People's Party, Kemal Kilicdaroglu, chimed in, saying: "We want the imperialist powers to leave the Middle East."

Turkey is one of the "triad" of governments -- with Russia and Iran -- that have assumed the task of bringing peace, or at least the end of conflict, to Syria. But the Afrin operation could upend that goal, and allow jihadist groups in Idlib, as well as ISIS, the opportunity to regroup.

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ISIS fighters made a sudden reappearance in south Idlib last week, killing and capturing a number of regime soldiers. It later published photographs of the execution of three men it said were regime soldiers.

After a gradual reduction in violence in Syria last year, and the elimination of ISIS' hold on Raqqa and Deir Ezzour, the new fighting in the north of Syria could quickly ignite a broader conflict, dragging in multiple factions.

According to the United Nations, more than 200,000 civilians have fled the regime's offensive in southern Idlib since mid-December, joining more than a million displaced Syrians penned into an ever shrinking patch of rebel-held territory where the former al Qaeda affiliate Hayat Tahrir al Sham holds sway.

Any thought those desperate civilians may have had of seeking refuge in Kurdish-held Afrin, already crowded with an estimated 600,000 people, is fast evaporating.