Meanwhile, the US commander in northern Iraq said he does not have enough troops for the mission in increasingly violent Diyala province, north of Baghdad.
He also said that Iraqi government officials are not moving fast enough to provide the "most powerful weapon" against fighters - a government that works and supplies services for the people.
Describing Friday's attacks, an Iraqi army source said a truck bomb hit a bridge near the town of Taji, north of Baghdad, on the main highway connecting the capital with cities in the north.
He said it was quickly followed by a car bomb that killed four soldiers there.
Munzir Baig, Muscat, Oman
Send us your views
The US military said the bridge was impassable for northbound traffic.
A police source said eight policemen were among the dead in the attacks on the bridges south of Baghdad, but it was unclear how many casualties were caused by each blast.
Police said the first bomber damaged the old Diyala bridge.
Minutes later, a few kilometres away, another attacker detonated a truck bomb on the new Diyala bridge.
The two bridges over the Diyala river, a tributary of the Tigris, are commonly used by Shia pilgrims on their way to holy Shia cities to the south.
Last month, a truck bomb destroyed the Sarafiya bridge in Baghdad.
On Friday, Major-General Benjamin R Mixon told Pentagon reporters by video conference from Iraq: "I do not have enough soldiers right now in Diyala to get that security situation moving.
"We have plans to put additional forces in that region."
Mixon commands the area that includes Diyala.
He said he has already received extra troops, but violence in Diyala is on the rise both because more fighters have moved in and because multinational forces are taking the offensive.
"We have made progress ... we have taken terrain back from the enemy.
"We are sure there are elements of both Sunni extremists and Shia extremists that have moved out of Baghdad and relocated into not only Diyala province, but also into Salah ad Din province."