BEIRUT, Lebanon — They seek to sweep away borders and establish a stateless society. Their female fighters struggle and die beside male comrades. Their leftist, anti-Islamist image has attracted American and European volunteers.
The Kurdish fighters who are battling the Islamic State jihadists in Syria are regarded by the United States as its most reliable partners there. But to Turkey, a NATO ally of the United States, these Kurds are terrorists.
The Kurdish group, known as the People’s Protection Units, or Y.P.G., is now facing an escalating battle with Turkish forces in northwestern Syria, complicating American policy.
The group has deep ties to the Kurdistan Workers’ Party, also known as the P.K.K. Both Turkey and the United States consider the P.K.K. to be a terrorist organization for its violent separatist movement inside Turkey.
While Y.P.G. leaders play down their P.K.K. ties, areas they control are festooned with photos of the imprisoned P.K.K. leader, Abdullah Ocalan, viewed by Turks the same way Americans viewed Osama bin Laden.
One thing is clear: The United States, which has relied heavily on Kurdish fighters to push the Islamic State out of northeastern Syria, has consistently understated the complexities of its alliance with the Kurds, a policy some analysts call willful ignorance.
“Obviously the U.S. chose to look the other way, out of what it deemed to be the necessity of building an alliance to quickly capture territory from Daesh,” said Noah Bonsey, a Syria analyst with the International Crisis Group, using the Arabic acronym of the Islamic State.
“The U.S. has sound reasons to continue to support the Y.P.G.,” he said, “but doing so while the P.K.K. maintains an active insurgency against its NATO ally is an unsustainable situation.”
The United States military’s official partner in Syria is a militia known as the Syrian Democratic Forces, which includes Arab and Assyrian fighters but is dominated by the Y.P.G. The Americans de-emphasize such details.
But the cooperation with the Y.P.G., including arming and training the fighters and providing them with air support, has put the United States on a collision course with Turkey.
Last weekend Turkey launched a military operation against the Kurdish enclave of Afrin in northwestern Syria and it is now threatening to expand the operation to the east, into areas where Kurdish forces are directly backed by the American military.
The United States has sought to differentiate between the Kurds it supports and those in Afrin, whom it does not, a distinction the Kurds themselves do not recognize.
“They are not different parts at all, and they cannot be divided in any way, not politically, not economically, not militarily,” said Newaf Xelil, a Kurdish political analyst in Germany and a former spokesman for the party affiliated with the Y.P.G. “For us, it is all Kurdistan and we are now defending Afrin with all we have.”
On Wednesday, President Trump urged his Turkish counterpart, Recep Tayyip Erdogan, to use restraint and avoid any situation that could lead to clashes between the Turkish and American militaries. Disputing the White House’s description of the call, Turkish officials denied that Mr. Trump had made such a request.
Y.P.G. leaders say theirs is a homegrown movement that sprang up to defend civilians in the early days of Syria’s war and against offensives by the Islamic State.
That role, and the backing of the United States, has transformed the group into the most prominent political and military force in northeastern Syria.
Formerly an impoverished and marginalized minority, Syria’s Kurds now administer substantial territory, where they are teaching Kurdish in schools and setting up local administrations. Critics have accused them of displacing Arabs.
American officials have long sought to minimize the Y.P.G.’s ties to the P.K.K., but Turkey is enraged that the United States is giving military support to a group that idealizes Mr. Ocalan, the sole inmate of an island prison in the Sea of Marmara.
Many Y.P.G. leaders speak openly of their history with the P.K.K., and Kurds from Iraq, Iran and Turkey have joined the movement in Syria.
Mr. Bonsey said there had been hope among the Americans that they could pull the Y.P.G. away from the P.K.K.
But such a prospect appears unlikely — especially with the Kurds now uncertain that they have solid support from the United States, which has sent mixed messages about how strongly it would back them against a Turkish onslaught.
The American ambivalence was clear on Wednesday in comments by Thomas P. Bossert, Mr. Trump’s homeland security and counterterrorism adviser, at the World Economic Forum in Davos, Switzerland.
“I’m not in any way critical of the Turkish decision, but I’m just praying for their longer-term strategic patience,” Mr. Bossert told reporters.
Asked if the Turks should withdraw, Mr. Bossert said, “I would prefer it if for now they would remove themselves from the capital of Afrin.”
The United States effectively gave a green light to the current Turkish offensive against Afrin, urging restraint but emphasizing that it does not work with the Y.P.G. there.
The enclave is in northwest Syria, not connected to a larger territory held by the Syrian Democratic Forces in the country’s northeast, where several small American military bases and several thousand American advisers are.
But Mr. Erdogan has threatened to attack that larger area, beginning with the town of Manbij. The Turks say the Americans promised that the Syrian Democratic Forces would withdraw from such majority-Arab areas after taking them from the Islamic State, but it has not.
The ingredients for this clash have been brewing since Syrians rose up against the authoritarian rule of President Bashar al-Assad in 2011. Within a few years, the northeastern province of Hasaka, with a large Kurdish population as well as Arabs and Assyrians, was effectively ruling itself.
As it became the area’s dominant force, the Y.P.G. tried to implement its vision of a utopian society, inspired by Mr. Ocalan. Influenced by Murray Bookchin, an American anarchist, Mr. Ocalan has called for autonomous rule by local committees unbound by national borders. Proponents say they do not seek to break up Syria but are leading a long-term social revolution that will ensure gender and minority rights.
When the Islamic State began sweeping across northeastern Syria, its first defeat was in the Kurdish border town of Kobani. The Y.P.G., with help from American airstrikes, repulsed Islamic State fighters at great cost.
After that, facing Turkish criticism, the United States began calling its partner the Syrian Democratic Forces. There was a scramble to enlist Arabs and members of other ethnic groups.
The Y.P.G. remains the predominant part of the Syrian Democratic Forces and provides its most effective fighters. But the Kurds do have local allies.
They have built local councils with representatives from various ethnic groups, each required to have a chairman and a chairwoman. While they are still largely overseen by the Y.P.G., locals have begun to engage with them in significant ways.
Hassan Hassan, a Syrian analyst and author, said that the Americans repeatedly had broken promises to Turkey about the alliance: They said the Syrian Democratic Forces would not cross the Euphrates River. It did. They said it would withdraw. It stayed.
The Americans also said that the Y.P.G. would refrain from promoting P.K.K. ideology outside Kurdish areas. But Y.P.G. fighters hoisted a huge poster of Mr. Ocalan in the center of the city of Raqqa after wresting it from the Islamic State last year, a huge embarrassment to the United States.
Mr. Hassan said the local councils are vetted by the Y.P.G. and cannot be seen as fully democratic. But he said that in some areas, Arab residents nonetheless had welcomed the Syrian Democratic Forces, particularly in the eastern province of Deir al-Zour, the area of Mr. Hassan’s family roots.
American officials have told the Kurds that the United States will not fight Turkey for them. With a diminished need to fight the Islamic State in Syria, Mr. Hassan said, American support for the Kurds could dwindle.
But Mr. Xelil, the Kurdish analyst, said the Americans would need the Kurds for their other objectives in Syria, like pushing back on Iranian influence.
“We think that it is surely possible that the Americans will find real reasons to deepen their relationship with the Kurds in a strategic sense,” he said.