The battle to retake west Mosul began, for me, rattling around in an armoured Humvee with two Abaases. “I’m Abaas One. He’s Abaas Two,” the driver, Abaas Almsebawy, said in English with a broad smile, pointing to the gunner on top.

“I have killed two Da’esh,” Abaas Two said, using an Arabic acronym for the so-called Islamic State (IS). “Well, one for sure. The other one crawled away but he was bleeding badly. I was told he died.”

Abaas One was jealous of his gunner’s luck. He was shot twice by IS in the city of Ramadi, in central Iraq; he still had a bullet lodged in his back. “The doctor said it is my gift from Da’esh,” he told me and laughed.

Over the sound of gunfire and mortars, the two Abaases called out to each other, giving directions, spotting targets. The cry of “Abaaaaas!” was constantly in the air. One from Babylon, the other from Baghdad, they stretched out on a felt blanket inside the armoured vehicle during lulls in the fighting and fell asleep, oblivious to its discomforts and the IS mortars landing outside.

They had been involved in the fighting in the east of the city, which it had taken 100 days to recapture, in hard, street-by-street clashes and through an onslaught of IS car and truck bombs. Yet the battle to retake the west, which began on Sunday 19 February and is being led by Iraq’s Emergency Response Division (ERD) and counterterrorism forces, has proved different – and faster.

Abaas One, the driver, was exhilarated. As Iraqi army helicopters flew overhead and the air force strafed villages with machine-gun fire and rockets, he rolled on, part of an armoured assault on a front that stretched for miles. His Humvee was built for this kind of terrain, moving at speed across the desert towards villages, the airport and eventually the city of Mosul.

Something else was different about this battle, too. These men were not technically soldiers: they were policemen. Abaas One went into battle in a hooded top and a leather jacket. Stuck outside manning his gun, Abaas Two, like a fighter from another age, wore a greatcoat, small, circular spectacles and a woolly hat. One lean and broad-shouldered, the other bulky and round-faced, they were a contrast but a good fit.

The Abaases were part of Iraq’s elite ERD, which has led the charge into the west of the city, just as the country’s heralded “Golden Division”, the counterterrorism unit, had pushed into the east. The ERD, part of the ministry of interior, is the less experienced junior brother of the battle-hardened Golden Division but it was determined that west Mosul would be its prize. It made swift progress and, as it took back village after village from IS, troops posed for selfies with enemy corpses on the roadside.

The closer to Mosul you were, the more charred bodies you would see, lying along the route. Two in a ditch, killed by a mortar, and two on the road, the motorcycle they were travelling on cut in half by an air strike.

In command of the 1st Brigade was Colonel Falah al-Wabdan. In Ramadi in 2015, he and his men had been cut off and surrounded by IS forces and had escaped only when more troops came to their rescue.

As he stood on the ruins of a former palace that had belonged to one of Saddam Hussein’s brothers, he had a view of all of Mosul. “I will be very glad when I see my forces move forward,” he said. “Also [when I see that] my soldiers are all safe. And I will be even happier when we have killed IS. These people [IS] are like a disease in the body, and we are now removing it, day after day.”

From there, the Iraqi forces took the town of Abu Saif, and then, in a six-hour battle, what was left of Mosul’s airport. Its runways were in ruins and its terminal buildings reduced to rubble. Yet that was the last open ground before they reached the city. By the end of the week, Colonel Falah’s forces had breached the IS defences. Now they were heading into the dense and narrow streets of the city’s old town. Meanwhile, the elite Golden Division was the secondary force, having earlier been bogged down in heavy fighting.

The competition between the two rival divisions had helped to accelerate the advance. The ERD, however, had a secret weapon. “We need to ask your men to hold off, sir. We have helicopters in the air,” the US special forces officer told an Iraqi lieutenant colonel on the rooftop as the assault on Abu Saif was in full force.

The Iraqi mortar team in the orchard and olive grove below held fire. Then the mighty thud of coalition air strikes could be heard and, just two miles away, a huge, grey cloud rose above the town.

***

It is Iraqis who are doing most of the fighting and the dying in the battle against IS, but since the Pentagon relaxed its rules of engagement late last year more Americans are at or near the front lines. They are calling in air strikes and laying down fire from their MRAP (“mine-resistant ambush-protected”) vehicles. They are not in uniform but, despite being a covert force, they are conspicuous and still wear the Stars and Stripes on their helmets. When journalists, especially cameramen, approach, they turn their backs.

In and around Mosul, it is more common now to get stuck in a traffic jam of US vehicles: either artillery or route-clearance teams. The Pentagon will soon respond to President Donald Trump’s call for a new plan – an intensification of US efforts against IS – but on the ground around this city, the Americans are already much more engaged in the fight against the militants.

British special forces were also in the area, in small numbers. Unlike their American counterparts, they went unseen.

Also seemingly absent in the early part of the offensive were civilians. It was three days before I met one: a shepherd, Ali Sultan Ali, who told me that he had only stayed behind because he could not get his flock to safety, as a nearby bridge had been destroyed.

As his sheep grazed, Ali explained: “They continued to attack this area, and now we are three days sitting in our homes, unable to go out because of attack and mortars . . . All the people, they have left this area one after another. They went to the east of the city of Mosul and they rented houses there because there are too many attacks here.”

Almost 60,000 people have fled west Mosul. In this area, with its population of three-quarters of a million, the battle has the potential to become a humanitarian crisis. Camps for internally displaced people still have capacity, but they are filling up.

IS, with anywhere between 500 and a few thousand fighters inside Mosul, is again using the local population as cover. But coalition air strikes may be taking a heavy toll on civilians, too. Officially, the US-led force claims that 21 civilians have died as a result of its bombs since November, but an independent monitoring group, Airwars, suggests that as many as 370 have been killed by Western aircraft since the start of March.

After the airport was recaptured, the columns of desperate people heading south began to thicken. The children among them usually held a white flag – perhaps a clever distraction thought up by terrified parents for their long walk to safety. Near the airport, I met a man who was too distraught to give his name. He told me that his brother’s family – six people – had been killed in an air strike. With his eyes red from crying and a blanket over his shoulders, he stood by the roadside, pleading. “For God’s sake,” he said. “We need you to help us. We need a shovel to get the dead bodies out of the building, because there are still two bodies under that building.”

But the battle was reaching a new pitch around him, so he left for a camp to look for his brother, the only remaining member of his family, he told me.

When the ERD finally made it inside the city, the first thing I noticed was the fresh laundry hanging in the yard of a family house. Then I heard a huge explosion as an IS truck bomb slammed into one of the Iraqi Abrams tanks.

The tank trundled on regardless and, by nightfall, the ERD had a tiny foothold inside the city: the al-Josak neighbourhood.

***

Islamic State is steadily losing Mosul and in Iraq, at least, the end of the so-called caliphate is in sight. In Abu Saif, state forces found the corpses of foreign fighters and, hiding, an IS operative who was still alive.

“He’s Russian,” one officer told me, but the man might have been from one of the central Asian republics. There were dead Syrians on the battlefield, too, men from Deir az-Zour; and for the tens of thousands of foreign fighters who joined IS, Syria will likely be a last refuge.

There may be another reason for the faster pace of the assault in west Mosul. The Iraqi forces, having fought IS in Ramadi, Fallujah and east Mosul, are getting better at dealing with the militant group’s tactics.

Truck bombs took a huge toll on their men in eastern Mosul. It is hard to describe the force unleashed when one of these detonates near you. In an early assault on one village, IS sent out four truck bombs and one of them exploded a few hundred metres from where I was standing. The shock wave ripped around the building and shards of engine went flying over our heads. My mouth was full of dirt. The debris was scattered for what seemed like miles around – yet no one died.

The suicide attack driver may have been taken out by an Iraqi soldier firing a rocket-propelled grenade (RPG). Whenever they advance now, men stand ready with RPGs, specifically to tackle the threat of car bombs. And they are becoming better at “hasty defence”. An armoured bulldozer is always in the lead. When a new street is taken, defensive berms made of mud or rubble are built to halt any speeding car bombs.

The IS fighters are crafty. Iraqi forces took me to a house on a captured street. Its yard was covered and the front wall was gone. Parked in the front room was what looked like an ambulance. Hidden from surveillance aircraft, this was another truck bomb.

“It’s still live. I wouldn’t go any further,” a major warned me. Even the bomb disposal team said that it was too dangerous to touch. It was later destroyed from a very safe distance.

Although the group violently suppresses modernity, IS fighters are innovators. They have no air force but they can get their hands on drones, which are commercially available, and they have “weaponised” them. If the battle for east Mosul was the attack of the car bombs, the battle for the west began as a drone war.

For the men on the ground, IS drones are enormously disconcerting. During a gun battle in west Mosul, I stopped to speak to some troops taking cover behind a wall. As I asked a final question, the captain I was talking to cupped his ear and leaned forward because of a sudden eruption of gunfire. Then, just to my right, I felt a shock wave of a detonation that seemed to come from nowhere.

A member of the BBC team was hit, receiving a small blast injury to the arm. When we got back to the Humvee, the driver explained that there had been a drone above us. The gunfire was from Iraqi troops trying to bring it down. The detonation had not come from nowhere; it had come from directly overhead. As we drove out of there, I noticed that the gunner had closed the hatch. We were protected inside, but he was outside manning his weapon, looking for more drones.

“They drop MK19 40mm grenades from the drones to stop the movements forward. All the time, they will use four to five drones to attack one location,” Captain Ali Razak Nama of the federal police explained. “As you know, we can’t always see these drones with our eyes, but if we do see them we can attack the drones with our rifles. [But] when we go into the battle, we are not looking at the skies. We are looking ahead of us for car bombs, suicide attackers, IEDs or snipers.”

A unit of the Golden Division was hit 70 times in a single day by wave upon wave of IS drones. The operator managed to drop a grenade inside a Humvee from above; all four men inside, members of a bomb disposal unit, were killed. Dozens more were injured that day.

The sound of a drone, even one of their own, is enough to make the Iraqi forces hit the dirt and scramble under a vehicle. They are difficult to bring down. I once watched as snipers and heavy machine-gunners opened fire on some drones; they managed to strike one but still it flew on.

The IS fighters control them from motorcycles in an attempt to prevent the operators being tracked and killed. They switch frequencies in the hope that they will not be jammed. Yet as a coalition commander told me: “The enemy aren’t going to win by dropping grenades from the sky. So it is certainly not a game-changer.” Iraqi and coalition forces now appear to be having success in countering the threat. Just how, they will not say, but in recent days there has been a “very significant” drop in their use.

***

Mosul has been the biggest battle for Iraqi forces against Islamic State, but commander after commander said that others had been tougher. In Ramadi and in Fallujah, IS had a better grip. In Mosul, the local people have been quicker to turn away from the militants.

In the eastern part of the city, the bazaars are busy again and children have returned to school. Girls are receiving education for the first time in nearly three years, since IS captured the city. The so-called caliphate was declared on 29 June 2014 and, four days later the new “caliph” and IS leader, Abu Bakr al-Baghdadi, made his first and only filmed appearance, delivering a sermon at the city’s al-Nuri Mosque. Iraqi forces are now in sight of the mosque, with its Ottoman-era leaning minaret.

Mosul is Iraq’s second-largest city and has a cosmopolitan heritage, but Islamists had influence here for many years before IS arrived. As one Mosulawi told me, after neglect by the Iraqi capital, “There is discontent with Baghdad, not support for Isis.”

Al-Baghdadi is believed to have fled the city already. According to US and Iraqi commanders, he is hiding out in the desert. Shia militiamen and Iraqi army forces are attempting to seal off escape routes to the west, into Syria. Yet senior commanders accept that in a city Mosul’s size, it will be impossible to close all escape routes. Capturing al-Baghdadi is not a priority, they say.

There is also an acknowledgement that neither his death nor the loss of Mosul will be the end of Islamic State. But in Iraq, at least, it will destroy the caliphate.

Quentin Sommerville is the BBC’s Middle East correspondent