The top US military officer vowed in Baghdad on Monday that the Islamic State group will be defeated, as Iraqi forces pressed their largest operation yet against the jihadists.

Some 30,000 men have been involved in a week-old operation to recapture Tikrit, one of the jihadists' main hubs since they overran large parts of Iraq nine months ago.

But in a sign of the brutal lengths to which IS will go to maintain control, it killed 20 men in Iraq's northern province of Kirkuk and strung up more than a dozen bodies in public.

General Martin Dempsey's visit also coincided with the start of an offensive by Kurdish peshmerga forces in Kirkuk that further increases the pressure on the last IS strongholds east of the Tigris river.

"Daesh will be defeated," Dempsey, the chairman of the US joint chiefs of staff, vowed at a news conference, using an Arabic acronym for IS that the group rejects as derogatory.

The United States began air strikes against IS in August, the first by what is now a 60-nation coalition of mostly Western and Arab states supporting Baghdad's fightback.

Dempsey emphasised that strikes must "be very precise" to avoid "additional suffering", also saying that while the priority has been protecting people, it may also be possible to use air power to defend Iraqi heritage sites.

Iraq's tourism and antiquities minister, Adel Fahad al-Shershab, called on Sunday for the coalition to protect such sites from IS, after the jihadists smashed priceless artefacts at Mosul museum, bulldozed one ancient city and may have attacked a second.

- No carpet bombing -

During a visit to a French aircraft carrier in the Gulf taking part in the air campaign, Dempsey appealed for "strategic patience" in the fight against IS in Iraq and Syria.

"Carpet bombing through Iraq is not the answer," he said on Sunday.

Dempsey stressed that training Iraq's army, which imploded when IS attacked in June 2014, would take more time, as would initiatives to bring its Sunni Arab minority back into the fold.

"I do think it's going to require some strategic patience," he said, adding that "these underlying issues have to be resolved".

Iraqi soldiers, police and the increasingly influential paramilitary Popular Mobilisation units, which are dominated by Shiite militias, have been closing in on Tikrit.

On Sunday, those forces retook Albu Ajil village, where some Sunni tribesmen have been accused of involvement alongside IS in the June 2014 massacre of hundreds of mostly Shiite recruits from the nearby base of Speicher.

Hadi al-Ameri, leader of the Popular Mobilisation units, had described the Tikrit operation as an opportunity for revenge, sparking fears for Sunni civilians.

Shiite commanders have since toned down their language and publicly urged their men to exercise restraint.

Dozens of families displaced by the fighting fled to Samarra, the other main city in Salaheddin province, to receive aid and shelter in camps.

- IS executes 20 -

"I am a farmer, I left my sheep and my cows behind," said Atta Abu Alaa, a 50-year-old who had fled with 12 family members from a village near Albu Ajil.

"We did not have any relations with IS, we were oppressed," he said, adding that those who fled were held and interrogated by the Asaib Ahl al-Haq Shiite militia for a day before being released.

The militia's leader, Qais al-Khazali, urged his fighters on the front line Sunday not to do anything that could tarnish a victory over IS in Tikrit, where just a few hundred jihadists are holding out.

In the Kirkuk province town of Hawijah, IS executed 20 men who wanted to join the Popular Mobilisation units, a police intelligence officer and two local officials said on Monday.

The executions could not be independently confirmed, but gruesome photos posted online are evidence that they took place.

The pictures show the bodies of more than a dozen men -- said to be Popular Mobilisation members -- strung by their feet from light poles, what appears to be a communications or electricity tower, and under a massive sign featuring the IS flag and name.

Iraqi Kurdish forces that control other areas of Kirkuk launched an operation backed by coalition air support on Monday that aims to push IS back south and west of the provincial capital, piling further pressure on the jihadists.

Security official Westa Rasul said Kurdish forces suffered casualties when fighters entered a booby-trapped house.

"The bomb went off, killing three Iranian peshmerga and several of ours," he said, refusing to say how many Iraqi peshmerga were killed.