“Fake facts!” exclaimed a senior Iraqi official in exasperation as he pointed to photographs online allegedly showing the Iranian General Qasem Soleimani in Kirkuk, orchestrating the Iraqi government retaking of the city last month. He said that in reality the picture, tweeted by a Kurdish leader as evidence of Iranian hegemony, dates from 2014.

The greatest threat to the growing stability of Iraq is the differences between the US and Iran being fought out politically – and even militarily – in Iraq. The Iraqi Prime Minister Haider al-Abadi said in an interview with the Independent earlier this week that his greatest concern is a US-Iran crisis. He added that “it is not my job to solve their differences, but it is my job to prevent their confrontation inside Iraq.” He hoped that mutual denunciations by Washington and Tehran would turn out to be rhetorical.

Given US hostility to Iran, the Baghdad government is alarmed by what it sees as an attempt to portray it as an Iranian proxy manipulated by General Soleimani and reliant on the Shia paramilitary Hashd al-Shaabi or Popular Mobilisation Units (PMUs). “Today’s offensive by Iraq, PMU Shia militia commanded by Iranian IRGC [Islamic Revolutionary Guard Corps] on Kirkuk have sadly started a new war in Iraq & Kurdistan,” reads a tweet from Hoshyar Zebari, a Kurdish leader and former Iraqi foreign minister last month.

The senior Iraqi official said that Gen Soleimani never meets the Iraqi Prime Minister Haider al-Abadi or anybody else of real importance in Baghdad and has also failed to get an audience with the Shia supreme religious authority, Ali Sistani, in the holy city of Najaf. He said: “In fact, Iranian influence over the Hashd has been going down over the last two years because they are no longer paying most of the groups, aside from Ketaeb Hezbollah.”

The propaganda war is intense and unscrupulous with Kurdish leaders in Irbil and much of the Arab media claiming that Iran pulls the strings in Baghdad, though the US is the government’s main military ally. The PMUs are portrayed as sectarian death squads which are leading the offensive into Iraqi Kurdistan. One video posted online purports to show the Kurds blowing up a bridge over the Lesser Zaab river at Altun Kupri, where Kurdish and Iraqi forces confront each other, to block the PMUs advancing into the Kurdish heartlands. In reality, the bridge is still standing and the much-watched video is of an entirely different bridge in Topeka, Kansas being destroyed in a controlled explosion to make way for new construction.

Iraqi forces enter Kirkuk Show all 8 1 /8 Iraqi forces enter Kirkuk Iraqi forces enter Kirkuk Iraqi forces flash the sign for victory as they head towards the city of Kirkuk AFP/Getty Images Iraqi forces enter Kirkuk Iraqi people gather on the road as they welcome Iraqi security forces members REUTERS Iraqi forces enter Kirkuk Iraqi children wave to Iraqi forces as they arrive AFP/Getty Images Iraqi forces enter Kirkuk Iraqis wave to Iraqi forces as they arrive in the first neighbourhood on the southern outskirts of Kirkuk AFP/Getty Images Iraqi forces enter Kirkuk A burnt Kurdistan flag is seen in Kirkuk REUTERS Iraqi forces enter Kirkuk Iraqi forces advance towards the city of Kirkuk AFP/Getty Images Iraqi forces enter Kirkuk Iraqis wave to Iraqi forces AFP/Getty Images Iraqi forces enter Kirkuk A member of the Iraqi forces walks past a defaced Kurdish flag painted on concrete blocks AFP/Getty Images

The US has always been paranoid about Iranian influence in Iraq and tends to conflate Iraqi Shia fighting for their community and variant of Islam with proxies under the control of Iran. The US Secretary of State Rex Tillerson on a visit to Baghdad last month said the Hashd should “go home”, apparently believing that its members were IRGC fighters from Iran. Mr Abadi speaks out vigorously in defence of the Hashd, but says he is determined that they must be under strict government control.

The power of the Hashd has become more limited today than when they were created as a mass movement three years ago by a fatwa from Grand Ayatollah Sistani - though several paramilitary organisations like the Badr Organisation, Asaib Ahl al-Haq and Ketaeb Hezbollah have a much longer history. This was in June 2014 when the Iraqi army had lost Mosul to Isis and looked as if it would be unable to defend Baghdad.

The Hashd was central in defending the capital and in early counter-offensives against Isis, but has increasingly had a secondary role in military operations which are now led by the highly trained and experienced Counter-Terrorism Force. In the nine-month siege of Mosul, the Hashd occupied territory outside the city, but the assault was led by the CTF, Federal Police and Emergency Response Division. There were no Hashd units in Kirkuk city earlier this week, though they do have joint checkpoints with the army along the road back to Baghdad.

The Hashd, who are part of the Iraqi security forces and paid for by the state, are becoming less independent and less influenced by Iran because the Iraqi government is much more powerful than it used to be. But there is no doubt that Sunni and Kurds are frightened of them and they have a nasty reputation for sectarianism and criminality. For all their claims to be obedient to the state, there is an Iraqi saying that there are four givers of the law in Iraq: the government, the religious authorities, the tribes – and the Hashd.

Qais al-Khazali, 43, the leader of the Asaib Ahl al-Haq Shia paramilitary group, denies that it is under the control of Iran or is sectarian. Dressed in white turban and black robes, he answers questions swiftly and articulately, showing a moderation that feels out of keeping with his violent past. Once a lieutenant of the nationalist populist cleric Muqtada al-Sadr, from whom he split in 2004, he set up Asaib Ahl al-Haq which rapidly gained a reputation for ferocity and close links to Iran. Arrested by the British in 2007, he was released in exchange for a British hostage in 2010. Speaking in his office in Najaf in an interview with The Independent, he was keen to emphasise that his group were neither sectarian nor pawns of Iran. “It is one of their lies,” he says in response to the charge of sectarian killings. “There has been no sectarian cleansing. I am adamant - we did not bring in any Shia families to a Sunni area.”

He says that American forces should leave Iraq because they are no longer needed. “They don’t want to leave, but we can force them to,” he says. “We have experience in resistance. If there is a mandate from the Iraqi parliament and the Iraqi people, then we will stand up to them.” This would be the sort of nightmare envisaged by Mr Abadi, in which Iraqi Hashd - which the US believes are under Iranian direction - start killing American soldiers.

As for the role of Iran and Soleimani in the taking of Kirkuk, Mr Khazali says that they were supportive to Mr Abadi and Iraqi government forces. “The reason the Prime Minister gets the credit is because he galvanised a great force to show he was serious.” He says that what Soleimani did was to pass on to the Kurds that Mr Abadi really meant business and they would not be able to resist.

As for the future of the Hashd, he says that “in future it should be completely amalgamated with the Iraqi military and should not be involved in politics.” Much here will depend on whether or not there is a prolonged confrontation with the Kurds in northern Iraq, in which case Baghdad will continue to need large military forces, including the Hashd. As of Thursday, Baghdad is threatening to end a truce and take military action after the failure of talks about the central government taking control of the borders of Kurdistan.