SANLIURFA, Turkey — Syrian rebels who have joined Turkey’s cross-border offensive against a Kurdish enclave say they see the assault as a way to settle personal scores, and reports are mounting of looting and summary executions carried out by the Syrian fighters.

Backed by Turkish troops and warplanes, thousands of Syrian rebels are now tightening their grip on the Kurdish-controlled Afrin enclave in northern Syria as part of an offensive that Turkey has framed as a matter of its national security.

Turkey has been fighting a decades-old war against Kurdish militants at home, who are closely allied with the People’s Protection Units, or YPG, just across the border in Afrin.

But for the Syrian rebels, who are mostly Arabs, participation in the Turkish offensive is personal.

“This is about revenge,” said Waleed al-Mahal, a former rebel fighter who says he has sent dozens of his relatives to Turkish recruitment offices to join the push and hopes to be called up soon himself. “We’re taking our land back.”

His rhetoric echoes a video circulated online this week, apparently showing rebel fighters discussing plans to loot houses in Afrin. “An eye for an eye, a tooth for a tooth,” says a fighter who is apparently in charge.

Like Mahal, most of the Syrians taking part in Turkey’s operation originally took up arms against Syrian President Bashar al-Assad’s army after anti-government protests spiraled into armed insurrection in 2011. A scramble for territory has pitted the rebels against Kurdish fighters, and both sides have traded accusations of war crimes.

Kurdish forces have taken advantage of an unprecedented opportunity to throw off decades of state repression and establish autonomous areas across Syria’s north. Much of the territory was captured with U.S. support, after Washington backed a majority-Kurdish force to seize areas ruled by the Islamic State. 

Monitoring groups and local residents say that Kurdish forces razed a number of Arab-majority villages in the eastern province of Hasakah while clashing with the Islamic State, and sought to remove some Arab civilians they feared might harbor extremist fighters if allowed to stay behind. The extent of those abuses remains unclear. 

Every Syrian rebel fighter interviewed by The Washington Post in Syria and southern Turkey had a story to share, some of the tales describing in detail what happened to family members, others apparently recycled from acquaintances.

“They killed dozens of members of our extended family, bulldozed our homes and joked that they would plant potatoes there,” said Mahal, who is from Hasakah, sitting in a spartan living room in this southern Turkish city. Above his head a green, white and black rebel flag was pinned to a large portrait of Turkish President Recep Tayyip Erdogan. 

Mahal and almost a dozen rebel fighters said Turkey would probably support them in a future advance toward Hasakah. But Turkish officials have given few indications that they are eyeing territory that far east.

Kurdish-led forces in eastern Syria said Tuesday that they were withdrawing from the front line against the Islamic State to resist the Turkish offensive in the north of the country.

Amnesty International warned last week that indiscriminate Turkish attacks around ­Afrin have killed scores of civilians. Although the rebels insist they target only combatants, allegations of abuses are mounting among local residents and monitors.

Some of those abuses have been caught in grainy cellphone footage that was filmed by the rebels themselves. In one, the fighters are shown looting olive oil and chickens from an abandoned home. In another, the rebels appear to execute a man in civilian clothes at point-blank range, pumping bullets into his body when he falls to the ground.

The videos have sparked terror among Afrin residents as the offensive draws nearer. “Everyone has seen them. Is that what’s coming?” asked a former schoolteacher, speaking on the condition of anonymity out of fear for his safety.

Turkish Prime Minister Binali Yildirim said Monday that Afrin was “surrounded” after fighters attacked from the north, west and east. “We have cleared all areas near our borders of terror nests,” he said at a rally in the central Turkish province of Konya.

Sensitive to any suggestion that the Syrian rebels have turned their backs on earlier goals, such as toppling Assad, in favor of Turkey’s cause, rebel commanders and spokesmen insist that participation in Turkey’s operation benefits all sides.

“We have mutual goals in this period,” said Osama Akkari, a spokesman for the Almuntasir Billah group. “We haven’t given up on fighting the regime, but we are having to shift priorities for now. No one is making us do this.”

Another rebel, speaking on the condition of anonymity because rank-and-file fighters are not permitted to speak with the media, laid out the new progression of the battles ahead as he hoped they would materialize. “Afrin, Manbij, Raqqa, Qamishli, Damascus,” he said, naming a string of Kurdish-held cities moving eastward through Syria before the government-held capital.

Last week, an influential Islamic council backed the Afrin fight, prompting fears that the abuses might escalate as the offensive reaches terrain where more civilians are hiding. Until now, the land the rebels have taken has been sparsely populated. The Syrian Islamic Council, a Turkey-based group, described Syria’s Kurdish fighters as “a corrupt group that aspires to destroy the earth.”

“To fight them is to wage jihad as long as they continue to champion the regime and are enemies of the revolution,” said the council in a statement approving the seizing of weapons and looting of Kurdish fighters’ possessions. 

In one video clip widely circulated online and among Afrin residents, a band of young fighters overlooks Afrin’s green plains, singing an Islamist militant song linking their offensive to those of extremists. “We resisted in Grozny, we resisted in Dagestan, we took Tora Bora as a den, our glory will rise in Afrin,” they sing.

Reached by phone, one of the men, Abu Zahra Al-Deiri, expressed bemusement at the attention his video was receiving. 

“I don’t know why people are making a big deal out of it — it’s just a song like any other,” he said. “They can’t really think we’re extremists, can they?”