Major General Tim Cross said he would not "want to be too optimistic" after Iraqi forces confirmed they had pushed Islamic State militants out of some agricultural areas outside the key city of Fallujah.
"IS is still a real threat, there's obviously a lot of trouble in Syria," he said on Premier's News Hour.
The city, located about 40 miles west of Baghdad, has been under Islamic State control since January 2014 and is considered the main supply line for the militants.
Iraqi Prime Minister Haider al-Abadi announced the beginning of military operations in a televised speech late on Sunday night. He vowed to "tear up the black banners of strangers who usurped this city" and hoist the Iraqi flag.
It comes after other advances in parts of Iraq and reports Islamic State had released a statement in Arabic saying the caliphate was no longer about land, a stark contrast to their previous proclamations.
Major General Cross said: "The air campaign has definitely made a difference, has definitely dented them, but most significantly the Iraqi military has been trained a bit better, they have been given additional equipment.
"They're now more confident and are beginning to put in these attacks".
But he added the threat from IS was far from over and said the end of the war against the group was still a long way off.
"They're making advances in Libya and they've got other players around the world still declaring themselves as part of the caliphate," he stressed.
Premier's Alex Williams speaking to Major General Tim Cross: