More than 10 people were reported injured in the attacks on Friday.
The fighting occurred during an organised parade of about 2000 members of al-Sadr's Jaish al-Mahdi (al-Mahdi Army) in Nasiriya, about 320km southeast of Baghdad.
An Associated Press reporter on the scene said the al-Sadr supporters, many holding the Quran above their heads, were marching towards the anti-US cleric's office, which is near the headquarters of the Dhi Qar provincial governor, Aziz Abid Alwan.
Armed men guarding the headquarters opened fire towards the crowd in an apparent bid to disperse it, prompting retaliatory shooting from al-Sadr supporters.
An official from Nasiriya General Hospital said four police officers and four civilians were wounded in the firefight and were being treated.
Another six al-Sadr supporters were wounded in the fighting, said Shaikh al-Khafaji, an official at al-Sadr's Nasiriya office.
The parade was attended by Fatah al-Shaikh, a member of Iraq's National Assembly.