Most of the more than 20,000 people who have reached safety since Iraqi forces launched an offensive last month are from the outskirts of the city, which lies only 50 kilometres (30 miles) west of Baghdad.
The 49-year-old man and his family were able to leave Fallujah this week but tens of thousands more civilians remained trapped in the city by IS.
This is the account he gave AFP by telephone of life under the jihadists and how his family eventually got out:
"We did not flee Fallujah when Daesh (IS) took over at the end of 2013," he said, referring to the start of a period of anti-government protests during which IS's previous incarnation gradually took over the city.
"We expected the crisis would end within weeks or months. But the gunmen soon had the people on a tight leash, imposing new rules, issuing decrees, setting up barriers and planting bombs on the streets.
"That continued through 2014 and 2015... We were already affected by this but our living conditions deteriorated abruptly at the beginning of this year.
"My neighbour and I contacted somebody called Abu Omar, who is a Daesh member known as the "wali" (local chief) of southern Fallujah, to facilitate our exfiltration from the city in return for smuggling his wife with us," Abu Marwan said.
"He had cut a deal for his wife to be taken to Kirkuk.
"Daesh calls the wives of its members 'State women' and women in the city who have not pledged allegiance to the movement are called 'common women'," Abu Marwan explained.
The deal was sealed with the 'wali' in two days. Abu Marwan prepared his car and his family and Abu Omar's wife all squeezed in.
"We were stopped at many checkpoints but when we said Abu Omar sent us, they let us through. Then he appeared on a motorcycle and drove in front of us to open the way. He told us we should keep a distance of 100 metres.
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Abu Marwan said they snaked their way down to an area called Zoba by the Euphrates.
"There were many Daesh fighters along the way, heavily armed and hiding in various shelters," he said.
A senior commander in the operation to retake Fallujah, one of the jihadists' most emblematic bastions, said up to 2,500 IS fighters were defending the city.
In Zoba, IS was out in numbers, controlling who left the area.
"Some families had been there for four days, waiting to cross the river. Some were arguing with Daesh," Abu Marwan said.
"I left my car with Daesh. After also arguing with them, we boarded a small boat with my family and Abu Omar's wife but Daesh said the men should swim," he said.
According to the Norwegian Refugee Council, which runs most of the camps where the people displaced by the operation are housed, hundreds of families fled through Zoba in recent days.
Several people were shot by IS trying to cross the river while others drowned, according to several aid groups.
SWEETS AND WATER
"After crossing the river, we walked a short distance and found the Iraqi army and the Hashed al-Shaabi" paramilitary umbrella organisation, he said.
"They welcomed us and handed out sweets, juice and water. Then they separated the men from the women, started conducting security checks and inspected our bags.
Abu Marwan said they were asked whether they had any information about IS leaders inside the city.
"The security forces began releasing the older men... The middle-aged and young men were kept all night as they carried out checks on us with laptops.
"I told them Abu Omar's wife had been with us but I have no information on what happened to her," said Abu Marwan, who was eventually allowed to leave and join his family.