AL-QADARIF, Sudan — From makeshift shelters in refugee camps here, just across the border from Ethiopia, refugees recounted what they saw while fleeing an escalating conflict in Ethiopia's northern Tigray region.

“They’re killing people madly,” said Zam Zam Maconin, 26, referring to government soldiers, her voice still filled with terror as she gripped her 5-year-old daughter. “We saw a lot of dead people on the way.

“We didn’t bring any food or clothes — we just escaped to save our lives and our children’s lives,” she added.

Nearby, Semere Tesfai described seeing human heads detached from bodies stabbed with knives and axes, terrifying the thousands who walked for days to seek safety in Sudan.

“This is a genocide against Tigray people,” said Semere, 35, an English teacher who said he fled the Tigray conflict last week along with cousins, neighbors and some colleagues. “It couldn’t be described as less than that.”

As he and relatives fled, Semere said, government soldiers shouted: “Abiy Ahmed rules. We will rule you.”

Civilians are fleeing fighting between Ethiopia’s military and forces loyal to the powerful faction that rules Tigray. Each claims that the other is the aggressor in an outbreak of violence that has already killed hundreds and that threatens to undermine economic and political reforms in the country and roil the volatile Horn of Africa.

Zam Zam and Semere’s accounts of the violence could not be independently verified. Billene Seyoum, a spokeswoman for Ethiopian Prime Minister Abiy Ahmed, denied that government troops were killing citizens and said they were targeting militia fighting on behalf of Tigray’s rulers.

“The guarantee we have from the national defense forces and the information we have is there is no attacking of civilians,” she said. “We are convinced that this has been something that is being deliberately manipulated, to make sure there is an influx of refugees to the other side, so as to escalate the call for humanitarian intervention and to internationalize the issue.”

Abiy pledged Wednesday that his government would protect ­Tigrayans fleeing the fighting.

“We vow to our innocent civilians that have fled, to protect their property, enable humanitarian support by the ENDF and guarantees their peace upon return,” Abiy said in a tweet, referring to the Ethiopian National Defense Force.

The U.N. Refugee Agency warned that a “full-scale humanitarian crisis” was unfolding around Tigray, with as many as 4,000 Ethiopians fleeing across the border into Sudan each day. A wave of 27,000 refugees has already poured into Sudan since the conflict erupted this month, and humanitarian groups on the ground were overwhelmed, the agency said.

“People are coming out of Ethiopia really scared, afraid, with stories saying they have been fleeing heavy fighting,” said agency spokesman Babar Baloch. “There is no sign of the fighting stopping.”

Abiy ordered troops into Tigray on Nov. 4 after the Tigray People’s Liberation Front (TPLF) attacked federal military bases and attempted to access artillery and military equipment. He has demanded that the TPLF, which for decades was the dominant political force in the country’s multiethnic ruling coalition, hand over control and ordered Tigray’s security forces to lay down their arms.

He has ignored growing international calls for negotiations to end the fighting, saying he will negotiate only when the rule of law is restored in Tigray.

The violence spilled across Ethiopia’s northern border over the weekend when Tigrayan forces launched rockets at Eritrea’s capital, Asmara, blaming it for siding with Addis Ababa.

U.S. Secretary of State Mike Pompeo in a statement Tuesday blamed the TPLF for what he called “its blatant attempt” to cause regional instability and condemned the missile attacks on Eritrea.

He urged all sides to show restraint, protect civilians and open up humanitarian corridors so that aid workers could help those in need. “Civilians, including U.S. citizens, should be protected from harm and be provided with humanitarian assistance and safe passage out of the conflict,” Pompeo said.

Abiy, who last year won the Nobel Peace Prize, warned Tuesday that federal troops were closing in on Tigray’s capital, Mekele. “The final critical act of law enforcement will be done in coming days,” he said.

Ethiopia says its air force struck unspecified TPLF targets outside Mekele on Monday and “liberated” four areas on the eastern front.

U.N. rights chief Michelle Bachelet expressed concern that the war could spiral out of control and raised the possibility that war crimes may have been committed. Amnesty International, citing witnesses, said last week that it had evidence of a massacre of civilians stabbed and hacked to death in the town of Mai-Kadra in Tigray, days after the conflict first flared.

At the temporary camp guarded by the Sudanese army, Zam Zam said food and blankets are running low amid the surge of refugees. She and others are waiting to be moved to U.N.-run camps where they will be settled.

“I don’t want anything but to see an end of the war so we can go back to our homeland,” she said.

Wroughton reported from Cape Town.