About 15 minutes later, the second car bomb exploded about 150 meters (yards) away, targeting people gathered near the scene of the first blast, Al-Issawi said. That blast killed four civilians and injured 20, including two policemen.
The target was unclear, but both cars exploded near a school and market.
Fallujah is located 40 miles (65 kilometers) west of Baghdad in the Sunni-dominated Anbar province. It was once a capital for Iraq's insurgency.
Earlier Monday, four civilians were killed and 11 others wounded when a minibus hit a roadside bomb in southeastern Baghdad, officials said. Among the dead was a 10-year girl.
The casualties were confirmed by two police officers and a hospital doctor, all of whom spoke on condition of anonymity because they are not authorized to speak to the media.
Even though insurgents have been weakened across Iraq in the last few years, deadly bombings and shootings still happen almost every day.