The deadly blasts, which sent smoke billowing over the Shiite Zafaraniyah area of southeastern Baghdad, came amid new U.S. efforts to break up insurgent cells in the capital responsible for such attacks.
The U.S. military announced earlier Friday that it had conducted a series of raids against car bombing networks across the country, killing four suspected insurgents and detaining nine others.
U.S.-led forces have focused on disrupting car bomb making factories after several high-profile attacks that have killed hundreds in Baghdad and surrounding areas in recent weeks.
Officials say al Qaeda-linked Sunni insurgents were trying to provoke retaliatory violence from mainly Shiite militias that had agreed to lay low to avoid confrontations with Americans during a 12-week-old security crackdown.
The twin attacks began about 6 p.m. when the driver of a sedan waiting in a line of cars outside a police checkpoint near the old Diyala Bridge, blew up his vehicle, partially collapsing the span, police said.
About two minutes later the driver of a large fuel truck barreled toward a second checkpoint at the nearby new Diyala Bridge and blew up his vehicle, police said. The bridge was also damaged, and firefighters worked to extinguish burning police and civilian cars that had been driving across during the attack.
Local resident Khalid Ahmed, who was waiting in line to cross the bridge during the first attack, was wounded in his shoulder and hand.
"I was four cars behind the car bomb that exploded and caught fire. I fainted and I opened my eyes in the hospital," he said.
The bombings at the bridges, which cross the Diyala River, a Tigris tributary, killed 23 people, including 11 police and 12 civilians, and wounded 57 others, including 26 police, police said.
Baghdad's bridges repeatedly have been targeted by bombers. The most serious attack occurred April 12 when a suicide truck bomb collapsed the steel-girder Sarafiyah bridge, plunging cars into the water and killing 11 people. Two days later, a suicide car bomb killed 10 people at the Jadriyah bridge.
In other developments:
The blasts Friday occurred despite a series of measures aimed at reducing violence in the capital.
U.S. and security forces have increased the checkpoints in the capital and, in a bid to prevent bridge attacks, banned trucks capable of carrying more than one metric ton from crossing without strict searches. They also have long imposed a four-hour weekly driving ban during Friday prayers in Baghdad, but that ban ended on schedule three hours before the attacks.
In one raid early Friday, troops acting on intelligence obtained in previous operations approached a building near Taji, an air base 12 miles north of Baghdad, suspected of housing a car bombing cell responsible for attacks on Iraqi civilians and U.S.-led forces, the military said.
The troops came under fire from four armed men, whom they killed in a gun battle, the military said. One of those killed was suspected of being a leader of the cell with ties to al Qaeda in Iraq's top leaders, the military said.
U.S. forces have staged several raids in the area in recent weeks aimed at the terror network's leadership, including one in which they killed al Qaeda propagandist Muharib Abdul-Latif al-Jubouri earlier this month.
Forces also carried out raids in Baghdad and the northern city of Mosul on Thursday and Friday, detaining a total of nine people suspected of producing bombs and smuggling foreign fighters into the country to carry out attacks against U.S. troops, the military said.
Two U.S. soldiers were killed in separate bombing attacks, the military announced Friday.
One soldier from the Multinational Division-North was killed Thursday from an explosion during combat operations in Diyala province, northeast of Baghdad, the military said. The second soldier was killed in eastern Baghdad when a bomb exploded near his patrol, the military said.
The deaths raised to at least 3,385, the members of the U.S. military who have died since the Iraq war started in March 2003, according to an Associated Press count.