The deputy governor of Baghdad, Ziyad al-Zawbai, has been injured in a gun ambush on his car that killed three of his guards.
Three of Mr Zawbai's security detail were killed in the attack
A bodyguard and a civilian were also hurt in the attack in southern Baghdad.
In another incident, two civilians were killed and eight people were wounded by a car bomb in central Baghdad.
The attacks follow a spell of relative calm during Iraqi polls last week, hailed by the US president as a boost for democracy in the Middle East.
Mr Bush told US television audiences on Sunday that Iraq has become a strong ally in the "war on terror".
He rejected calls for a speedy withdrawal of US troops, warning that this would strengthen the very forces they were fighting.
Last week's Iraqi election, he said, "means that America has an ally of growing strength in the fight against terror".
Monday's attacks in Baghdad bring to an end a short period of relative calm during the election, for which several insurgent groups had called a ceasefire.
Tyres were burnt in the streets of Basra to protest fuel price rises
Baghdad's deputy governor and his bodyguard are being treated for their injuries in a hospital's intensive care unit, an official told Iraq's al-Sharqiyah television channel.
Monday's car bomb attack in central Baghdad is said to have targeted a local police chief but it is not known if he was among the eight people wounded in the attack.
The bombing took place outside a children's hospital.
In other news:
Government plans to raise fuel prices prompt street protests in southern Iraq.
The driver of Susanne Osthoff, the German archaeologist taken hostage in Iraq last month and freed over the weekend, has also been released, reports say.
An Iraqi insurgent group releases a video purporting to show the execution of Ronald Allen Schulz, a former US marine and security contractor kidnapped in Iraq.