Western support has helped Iraq's Kurdish troops regain ground and momentum against the Islamic State group, but retaking the northern jihadist hub of Mosul remains too big an ask for now.

On June 10, the jihadists who already controlled parts of Syria overran Mosul, Iraq's second city, and swept across the country's Sunni Arab heartland virtually unopposed.

When federal forces buckled as jihadists advanced, the Kurds seized long-coveted territory around Mosul. But in August, they suffered setbacks of their own when IS surged again.

Air support from the United States and France, arms deliveries and improved ties with Baghdad have allowed the Kurdish peshmerga fighters to regroup and take the initiative.

Speaking at an international conference in Paris last week, Foreign Minister Ibrahim al-Jaafari even predicted it would "not be too difficult" to retake Mosul.

However Roz Nuri Shaways, Iraq's newly-appointed finance minister and a peshmerga commander whose forces recaptured seven Christian villages from IS last week, admitted Mosul was too ambitious an objective in the short term.

"We need real cooperation from the international community" to reclaim Mosul, a city which had a population of close to two million before its capture by the jihadists, he said.

It was in Mosul, in July, that IS supremo Abu Bakr al-Baghdadi made his only public appearance since becoming the region's top jihadist, and the city is seen as the place where the war against IS in Iraq will be lost or won.

"I have felt so relieved since the news that air strikes in Syria and Iraq had started," Saad Mahmud, a 34-year-old Mosul taxi driver, told AFP.

"We have had a taste of the horror and hope that the raids will go all the way, to teach a lesson to anyone who is prepared to support those criminals," he said.

US and French jets have been carrying out air strikes nearly all around Mosul but they are powerless to smoke IS militants out of their urban hideouts.

"The peshmerga cannot take Arab Mosul, and Baghdad doesn't have the force to do it alone," said Kirk Sowell, the publisher of the Inside Iraqi Politics newsletter, adding: "US airpower is irrelevant to that battle."

- Months away -

The June debacle saw federal forces abandon a lot of US-supplied heavy weaponry to advancing jihadist forces, making any fightback all the harder.

"The peshmerga have never been that heavily equipped," said Anthony Cordesman, an expert on the country's security forces from the Centre for Strategic and International Studies.

"You need to provide them with enough armed vehicles, light armour, artillery, ammunition and other supplies," he said.

Retaking Mosul "is a major combat operation and at least a matter of months unless IS implodes for some internal reason. You need time to build and rebuild the Iraqi forces," he said.

Nuri al-Maliki stepping down as Iraqi prime minister last month went some way to easing relations between Baghdad and the Kurds but distrust remains high.

There is a wide consensus that for Iraq's Sunni Arab minority to stop hedging its bets and throw its lot in with the government, Maliki's successor Haider al-Abadi would have to give them substantial guarantees.

Abadi has made gestures of goodwill but the political answer to the jihadist takeover of swathes of the country has yet to materialise.

Obscure resistance groups reportedly organised by university students from Mosul with backing from some former army officers have been staging hit-and-run attacks against IS, but not enough to shake the group's grip on the city.

"Mosul could be the last area to be cleared in Nineveh province," said Ahmed Ali, from the Institute for the Study of War.

"The priority will likely be first oriented to rural areas, Mosul's urban nature will make it a monumental task to clear."