Amiriyat al-Fallujah, Iraq - As the Iraqi armed forces and allied militias continue their battle to take Fallujah from the Islamic State of Iraq and the Levant (ISIL, also known as ISIS), a growing number of civilians have said that they were tortured after escaping the besieged city.

In one tent in the Amiriyat al-Fallujah camp, which is hosting displaced people from Fallujah, three men claimed to have been tortured while held captive by sectarian militias affiliated with the Popular Mobilisation Forces.

The men, along with their families and many others, fled the al-Azraqiyah area, northwest of Fallujah, as the Iraqi military advanced towards their houses. Although they knew the road to reach the camp was not completely safe, they decided to leave out of fear of what might happen if they stayed.

"We knew that the militias might arrest or even kill us, but we had to leave," said Abu Muhammad, 57, who asked that his real name not be used. "Staying in the area meant sure death, whereas if we tried to leave, we might have the chance of surviving - or if not us, at least our families."

READ MORE - Fallujah crisis: 'We are being left to slow death'

Since 2014, a number of sectarian militias, collectively known as the Popular Mobilisation Forces or al-Hashd al-Shaabi, has played a major role in assisting the Iraqi armed forces in the fight against ISIL. But they have also been accused of perpetrating human rights abuses against civilians on sectarian basis.

Abu Muhammad said he experienced "days of hell" during his detention by the militia fighters. "We saw men in uniforms with army vehicles. We thought it was the Iraqi army that was approaching. We walked towards them holding white flags to make sure they know that we are civilians and unarmed," he explained.

The families soon discovered that the men in uniforms were not army soldiers, but militiamen who belong to the mobilisation forces. "They separated women and children from the men, and put everyone in houses taken from families who had fled as the military operation started," Abu Muhammad said. After two days, the women and children were moved to the Amiriyat al-Fallujah camp, while the men remained in detention.

They would put me face down, and start pulling the rope higher and higher. I felt that my limbs would be detached from my body. I lost consciousness several times during those days. Thamer Hassoun al-Shoukor, a tortured Fallujah resident

According to Abu Muhammad, this is when the agony began.

"The militants told us: 'We know that you have fled a place which is more like hell - al-Azraqiyah - but we will send you to hell again'."

Abu Muhammad said he was beaten repeatedly for six days. "They would beat us with water pipes; they would take turns to torture us. My hands were tied behind my back, and one of the militants sat on my chest after he got tired of beating me. He just threw himself on me. I felt my ribs breaking. I screamed in pain; I spat blood. I asked for water, but I was denied even that."

A week after their ordeal started, the Iraqi federal police arrived and moved the hundreds of detained men to the al-Mazraa military base, located east of Fallujah. They were held there for interrogation, then transferred to Amiriyat al-Fallujah.

Mahmoud al-Naji al-Shoukor, who also fled al-Azraqiyah, said he had "lost the will to live" as a result of his and his family's treatment at the hands of the sectarian militia.

"I was humiliated by my countrymen for reasons I do not comprehend until now. My two brothers, Ahmad and Hussein al-Naji al-Shoukor, along with my two cousins Muthanna and Ahmad al-Naji al-Shoukor were detained the same night. But we still have not found them; they are still missing. I think they are dead - so I have nothing left to lose."

Al Jazeera's Omar Al Saleh:

The battle for Fallujah Iraqi army commanders estimate it will take them about 48 hours to clear Fallujah. I think this is very optimistic. There are between 400 and 1,000 ISIL fighters stationed inside the city - including its most experienced. In the early hours of Monday morning, Iraqi forces began efforts to advance from three fronts - mainly from the south and northeast. Fighting is going to be tough for both sides. We understand that there is heavy air power provided by the US-led coalition and the Iraqi air force. Fallujah is very symbolic for the government. It's very close to the capital, Baghdad, which can be reached in a 30-minute drive. Fallujah was the first Iraqi city to fall to ISIL in 2014. It was also the main Sunni city that fought against the Americans when they occupied it in 2003. While detained, Mahmoud said he saw civilians being slaughtered "like chicken", and that he himself was very close to being executed. "They put some of us in a line on our knees, with our eyes blindfolded," he said. "They started taking us one by one. They killed several men before it was my turn. I could hear the men scream and beg the militia men to spare their lives - they were swearing that they never fought, and that they never joined ISIL - but that did not stop the militia fighters from killing them," Mahmoud told Al Jazeera.

"One of the fighters dragged me on the ground, saying that it is my turn to die. But at that moment I heard a voice saying that the killing has to stop, because the religious authority in al-Najaf [the religious leader of Iraq's Shia, Ayatollah Ali al-Sistani] had issued an order that was circulating among the militia men to stop the executions after the news of sectarian killings had been leaked."

Mahmoud said the militiamen told them that if it weren't for Sistani's orders, all of them would have been killed in a "blink of an eye".

Removing his shirt, Mahmoud showed the marks of torture on his body: Deep wounds from beating and flogging.

The third man in the tent was Thamer Hassoun Mohammad al-Shoukor. He was unable to easily move his arms: For six days, his hands were tied behind his back from the elbow, and his legs tied together from the ankles.

"They would put me face down, and start pulling the rope higher and higher. I felt that my limbs would be detached from my body. I lost consciousness several times during those days," Shoukor said.

One night, they heard the sound of a helicopter approaching. It was the Iraqi federal police. "If they did not show up, we could have been dead," he said.

Upon the men's arrival at the al-Mazraa military base, they were immediately given food and water, and several medical teams attended to those in urgent need of care.

"When we arrived, lots of tribal fighters from Hashed al-Anbar, which is a Sunni militia fighting with the government forces, started comforting us and telling us that our families are safe in Amiriyat al-Fallujah camp, and that we will be released once we receive medical care," Thamer said.

That same night, they received a visit from Yahia al-Ghazi, a member of parliament from the province. "We asked him about our families and about other men who were detained with us. All he had to say was that, 'we thank God that you are alive', and that our families were in the camp. But he didn't give us any information on the missing men and boys."

Sohaib al-Rawi, the governor of al-Anbar province where Fallujah is located, confirmed that civilians fleeing the city and surrounding areas had been tortured by some militias fighting alongside the Iraqi armed forces.

Rawi told Al Jazeera that "more than 49 civilians have lost their lives under torture". He added that 643 men were missing, with no information on whether they had been executed or are still in detention.

Although Rawi refused to give the name of the militia that carried out these acts, he said that "the government and the security forces were given the names of the 643 who are still missing".

Last week, Human Rights Watch issued a report stating that it had "received credible allegations of summary executions, beatings of unarmed men, enforced disappearances, and mutilation of corpses by government forces over the two weeks of fighting, mostly on the outskirts of the city of Fallujah".

Quoting witnesses who survived the killings, the report found that "a group consisting of Federal Police and PMF [Popular Mobilisation Forces] had separated men from women, marched the men to where the troops' officers were, lined them up, and shot at least 17 of them, including one teenage boy".

On Wednesday, June 15, Salama al-Khufaji, a member of the Iraqi Commission for Human Rights, announced that Iraqi security forces had arrested a militia fighter in connection with the killing of the 17 civilians.

However, it is yet to be seen whether other militiamen involved in abuses will face justice.