The Islamic State terror group has held the Iraqi city for more than two years. Iraqi forces have launched an offensive to retake Fallujah.
Civilians have been killed in heavy shelling or buried alive under the wreckage of their homes, according to reports received by the United Nations refugee agency, or UNHCR.
"There are also reports of several hundred families being used as human shields" by Islamic State militants, said agency spokesman William Spindler. Militants are putting those families in or around military targets in hopes of preventing an attack, he said.
About 625 families have escaped the fighting since last week.
Iraqi forces are separating men they capture from their families to ensure who is IS and who is not.
"Approximately 500 men and boys over 12 years old are held for security screening which can take five to seven days," Spindler said.
VOA spoke about the process with a refugee named Ayad.
IS militants “questioned me, the Hashd al-Shaabi and the Iraqi army,” he said. Hashd al-Shaabi is the name of the group of Shi'ite militias that work with the Iraqi military.
Ayad said Islamic State warned that “if we find your name … we will kill you. If you are in the clear, you can live with us in peace.”
Hashd al-Shaabi has been a leader in fighting IS. Some observers see the pro-Iran militias as better trained than the Iraqi army.
But others worry that a strong Shi'ite presence in the fighting will worsen religious divisions in the country.
Many Sunnis in Baghdad say they live in fear of the Shi'ite militias. Human rights organizations have noted abuses by the militias in the past.
Iraqi Lawmaker and former National Security Advisor Mowaffaq al-Rubaie defended the Hashd's role in the fight against Islamic State.
“I believe the Hashd al-Shaabi should have, and is having, a pivotal role” in the freedom of our occupied territories, Rubaie told VOA.
"The Hashd al-Shaabi fight the IS in a guerrilla-war style, not a traditional style, not like the Iraqi army," he said.
"But the Iraqi army, the federal police and even the anti-terrorism (forces), they fight in a traditional way. That is why the destruction is huge," he said.
The presence of the pro-Iranian Shi'ite militias also feeds into concerns over Iran's growing power in the Middle East.
Iranian General Qassim Suleimani visited the Fallujah area over the weekend, raising the concerns of Sunni leaders in Baghdad. Fallujah is traditionally under Sunni control.
Kareem Nouri, a spokesman for the Hashd al-Shaabi, defended the presence of Iranian advisors.
“We have American security advisors sitting in the Green Zone and we have Iranian security advisors on the front line helping us, but only with advice. Nothing else. The Iranian advisors are more helpful than the Americans," he told VOA.