"Reports from families that NRC has been in touch with describe that civilians trying to cross the Euphrates River in order to flee the fighting are being targeted by armed opposition groups," the organisation said in a statement.
NRC runs the camps in the town of Amriyat al-Fallujah to which most of the civilians who have fled areas around the besieged jihadist bastion are being housed.
"An unidentified number of civilians have been shot and killed trying to cross the river," NRC said.
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One of the only ways for civilians to try to leave the centre of Fallujah, which is littered with booby traps and roadside bombs, is to sneak out by river.
Most of those who have already reached camps lived in outlying areas and as many as 50,000 civilians are believed to remain in the centre of Fallujah, being used as human shields by IS.
"Our biggest fears are now tragically confirmed with civilians being directly targeted while trying to flee to safety," NRC country director Nasr Muflahi said.
"This is the worst that we feared would happen to innocent men, women and children who have had to leave everything behind in order to save their lives," he said.
The aid group said that around 18,000 civilians have reached displacement camps since Iraqi forces two weeks ago began an operation to retake Fallujah.