Falluja lies just 50km (30 miles) west of Baghdad and has been held by IS since 2014. It is one of two remaining IS strongholds in Iraq.
Iraq’s army began the fight to retake the city late last month.
Shakir al-Essawi, the head of Falluja’s regional council, told Reuters people were trying to cross the Euphrates in refrigerators, cupboards and barrels.
“Our biggest fears are now tragically confirmed with civilians being directly targeted while trying to flee to safety,” said Nasr Muflahi, the Iraq country director of the NRC, that said only that “armed opposition groups” were behind the shooting.
“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.”
Civilians fleeing Falluja, an Iraqi stronghold of the so-called Islamic State (IS), are being shot as they leave, an aid agency said.
The Norwegian Refugee Council (NRC) said families it had interviewed described people being shot dead as they crossed the Euphrates River.
The NRC, that runs refugee camps near Falluja, said up to 50,000 people remained in the city.
