The Norwegian Refugee Council said the 4,000 who fled since Saturday bring the total number of residents who escaped Fallujah since the offensive started in late May to 27,580.
Thousands more are expected to take the risky journey in the coming hours, NRC said, adding that some refugees reported IS militants were demanding 150,000 Iraqi Dinars, or around $130, from each person who wanted to leave.
The aid group warned that humanitarian resources are running low and called for at least $10 million to fund a six-month supply of water, food and basic necessities.
"Thousands of others remain trapped inside and the most vulnerable will need urgent assistance," said Nasr Muflahi, the NCR country director in Iraq.