"It would have taken us much, much longer to take this back, if it hadn't been for the air strikes," said the Peshmerga commander as he picked his way through a sea of rubble and debris.
He was at the unfinished hospital building in Rabia where militants of the Islamic State (IS) held out for three days against Kurdish forces as they battled to regain the strategic border town last week.
Rabia controls the main highway linking the two biggest cities of the northern Arab world, Mosul in northern Iraq and Aleppo in Syria, at the point where it crosses the border between the two countries.
The hospital building where IS fighters made their stand was devastated by the fighting. It had never been commissioned as a hospital, and will need to be completely rebuilt before it is.
While the basic structure is still standing, huge blast-holes show where missiles or bombs from coalition air strikes penetrated the building, blowing out much of the interior.
This was where the British RAF staged its first air strikes in Iraq in the current conflict, after conducting many reconnaissance flights.
Still sprawled in the chaotic debris are the gruesome remains of several IS militants who died there, their corpses left to rot in the rubble after a desultory attempt to cremate them by pouring fuel over them and setting them alight.
The battle for the hospital building illustrated the effectiveness of air strikes which, the Kurds say, sped up their operation and probably enabled them to avoid taking heavier casualties themselves.
But the limitations of Western coalition support for the Kurds were highlighted by events just 10km (six miles) away to the east, on the highway that leads to IS-held Tal Afar and Mosul.
On the day that Rabia fell, the Kurds say an earthworks barricade they erected on the road to prevent IS bringing up reinforcements came under attack from no fewer than seven suicide vehicle bombs.
Several of the attacks were foiled when Peshmerga fighters shot and killed bombers in civilian cars before they could set off their charges.
But a tanker truck with an armoured cab defied everything the Kurds had to throw at it.
It crashed into the barricade, hitting one of the unexploded car bombs and setting off a blast which devastated several Peshmerga vehicles.
Among the 10 members of the Peshmerga's elite special forces who died in the explosion was one of its commanders, General Shaikh Omar Babkai, who we had filmed just two weeks earlier on a visit to the front by the Kurdistan president, Masoud Barzani.
After the death of Gen Babkai, a veteran Peshmerga who was wounded five times over the years in battles with Saddam Hussein's forces, we drove to his home village, Mamola, in the mountainous far north of the country near the Turkish border, where we spoke to his brother Samad.
Like Gen Babkai and many other men in the area, Samad is also a Peshmerga officer. Gen Babkai's only son, Mir Khan, who is 17, is preparing to join the Peshmergas when he finishes his studies.
"We are proud of his martyrdom in this way, and we are ready to offer more sacrifices to protect and liberate Kurdistan," Samad said.
"But there is a deficiency in the arms we have. If they'd had more advanced weapons there, my brother would not have been killed, and the enemy would have been smashed much more quickly.
"Only the Peshmerga are seriously fighting IS," he added. "If we had the kind of arms the Iraqi army was given, we'd have destroyed the terrorists long ago."
In Mazne, a nearby village in this Kurdish heartland, the point was echoed by one of the survivors of the suicide attacks at Rabia.
Nabih Hassan was with three other Peshmergas, all from the same family in Mazne, who were in one of the vehicles destroyed in the bomb blast.
Badly wounded, he was the only one of the four who lived.
"If we'd had more sophisticated weapons, such as anti-armour missiles, we could have blown up the tanker while it was still far away, before it could get to us," he said.
One of the Mazne Peshmergas who died in the same vehicle was Ziro Hassan, who was manning an anti-aircraft "doshka" weapon used as a machine gun.
"He is not the first martyr we have given, nor will he be the last," said his brother Majid Hassan, also a Peshmerga general.
Such words, repeated by all the bereaved, are not empty rhetoric.
"Since 1961, our family has given 39 martyrs," he said. "We are ready to sacrifice to the last drop of our blood to preserve Kurdistan."
While the Kurdish readiness to fight and die for their country is well established - "Peshmerga" means, roughly, "those who defy death" - the IS cult of embracing "martyrdom" by committing suicide on the battlefield is something else.
We saw evidence of it at the same front-line position east of Rabia where the multiple vehicle bomb attacks were mounted.
Three days later, the IS militants launched a frontal attack on the Peshmerga defence line by around 100 fighters, many of them strapped up with suicide belts.
But the Kurds were ready. The maize and sunflower plantations on the plain in front of them were turned into a killing field. The Peshmerga said many of the attackers exploded as they were hit by gunfire.
We found at least 20 IS corpses scattered in the dirt. One was still wearing a suicide device, its detonator clearly visible. All that was left of another, whose explosives had gone off, was his head and upper chest.
"They are ready to die, they love death, and it's hard to stop an enemy if they want to die," said Masrour Barzani, head of security and intelligence in Iraqi Kurdistan.
"This is the enemy we are facing - they love to kill, they love to die, and unfortunately, they have access to the weapons they need to fight with.
"The problem is that right now the kind of support and armament that we are getting is not to the level where it can help the Peshmergas fight this enemy, especially when they have these armoured vehicles.
"We have not asked for any ground forces. Our Peshmergas are here, they are giving their lives, and all we need from the rest of the world is to help us with effective weapons to protect these people who are actually fighting on its behalf, fighting these terrorists who have come from all over the world."
Kurdish leaders say that, in addition to armour-piercing weapons which would stop suicide bombers and other armoured IS attacks, they believe that a qualitative upgrade to tanks and helicopters would give them a much better chance of defeating the militants.
They admit that such weapons would require extensive training, which would take time.
"But if we don't start now, we won't have them next year, when this war will still be going on and we will need them," said one senior official.
"It's going to be a very tough fight, and we're going to lose people," added Mr Barzani.
"We are not getting what we need. Air strikes are very effective, we're grateful for them and we hope that they could continue and be expanded, but definitely there is much more to be done, especially on the ground."
So far, most of the arms supplied to the Kurds since the current crisis began have been restricted to light weaponry and ammunition, though Britain has delivered 40 heavy machine guns and the Germans are training the Peshmerga on Milan anti-tank rockets.