"I saw a dead National Guard burning on the ground," said one witness after the first bombings.
"I saw a car trying to reach the town hall," said bookshop owner Mohammed Ahmed. "When police stopped it, it exploded."
Separately, police said insurgents shot dead another policeman and are suspected of firing a mortar that killed a woman and a young boy in a house near a US base in the city.
A Marine spokesman said an attack on a US convoy wounded 20 marines in Ramadi, 110 kilometres west of Baghdad. A police source said it had been a car bomb blast.
Hospital staff said at least one Iraqi was killed and 14 wounded in clashes between insurgents and US forces in Ramadi.
In Baghdad, a powerful explosion struck the main airport road, destroying what appeared to be a Humvee military vehicle, witnesses said, but there was no word on casualties.
On Saturday, the fiercest US air and artillery strikes on Falluja in months destroyed a hospital, a medical warehouse and dozens of houses, dazed residents said after a sleepless night.
Hospital staff said ambulances had been unable to go out as the city shook to explosions. Later they collected two dead and seven wounded civilians, among them women and children.
Insurgents also fought with US troops near a highway just north of Falluja and American planes bombed targets on the northern edge of the city, witnesses said.
Most of the city's 300,000 people have already fled. After Friday night's barrage, many more streamed out of the city to the Northwest on the only road left open by US forces.
UN Secretary-General Kofi Annan has warned that an attack on Falluja could undermine the elections, but his comments drew a chilly response from interim Prime Minister Iyad Allawi.
Allawi says Falluja is a haven for former Saddam fighters and militants led by Jordanian Abu Musab al-Zarqawi.
Zarqawi's group said it was behind a suicide car bombing on Thursday that killed three British troops south of Baghdad.