More than 20 protesters have been killed in a government crackdown on demonstrations in cities in southern Iraq.

Thousands defied a curfew, which was imposed in Najaf after demonstrators burned its Iranian consulate, to march in the funerals of those who had been killed.

Iraq's capital and south have been torn by the worst street unrest since the 2003 US-led invasion that toppled Saddam Hussein, as a protest movement has vented their fury at their government and its backers in neighbouring Iran.

Joint military-civilian "crisis cells" have been set up to try to stem unrest and a paramilitary commander vowed to use force to stop any attack against Shi'ite Muslim religious authorities.

The torching of the consulate in Najaf, the southern holy city, escalated violence in Iraq after weeks of mass demonstrations that aim to bring down a government seen as corrupt and backed by Tehran.

Demonstrators gather as Iran's consulate in the southern Iraqi Shiite holy city of Najaf burns

It was the strongest expression yet of the anti-Iranian sentiment of Iraqi demonstrators as the gulf widens between a largely Iran-aligned ruling elite and an increasingly desperate Iraqi majority with few opportunities and minimal state support.

The inability of Iraq's government and political class to deal with the unrest and answer protesters' demands has fuelled public anger.

Prime Minister Adel Abdul Mahdi has promised electoral and anti-corruption reform but barely begun delivering while security forces have shot dead hundreds of mostly peaceful demonstrators in the streets of Baghdad and southern cities.

Young, mostly Shi'ite protesters say politicians are corrupt, beholden to foreign powers - especially Iran - and they blame them for a failure to recover from years of conflict despite relative calm since the defeat of Islamic State in 2017.

Security forces opened fire on protesters who had gathered on a bridge in Nassiriya before dawn, medical sources said. Sixteen were killed and dozens wounded, they said.

"The burning of the consulate last night was a brave act and a reaction from the Iraqi people - we don't want the Iranians," said Ali, a protester in Najaf.

"There will be revenge from Iran I'm sure, they're still here and the security forces are going to keep shooting us."

A protester who witnessed the burning of the consulate said security forces had opened fire to try to stop it.

The military commander of the Popular Mobilisation Forces (PMF), an umbrella grouping of paramilitary groups whose most powerful factions are close to Tehran, said the groups would use full force against anyone trying to attack Iraq's most powerful Shi'ite cleric, who is based in Najaf.

"We will cut the hand of anyone trying to get near (Grand Ayatollah Ali) al-Sistani," commander Abu Mahdi al-Muhandis said in a statement on the PMF website.

Observers said the events in Najaf would likely bring a tough response, rather than pushing the government into enacting reforms.

"Apart from casual statements ... the government has not announced any plan (or) given any clear account of what measures it will take," said Dhiaa al-Asadi, advisor to powerful populist cleric Moqtada al-Sadr. "Initiatives are going to be scarce."