Iraqi forces struggled Wednesday to break into Fallujah city centre where hundreds of fighters from the Islamic State group and some 50,000 increasingly desperate civilians were holed up.
Fighting also raged hundreds of kilometres (miles) further up the Euphrates Valley in Syria, as US-backed Kurdish and Arab fighters opened a new front against the jihadists in the strategic Manbij pocket on the Turkish border.
Elite Iraqi forces positioned on the edge of Fallujah, one of IS's historical strongholds, met fierce resistance from besieged jihadists seemingly condemned to making a suicidal last stand.
"Our forces are still pushing to break into the city centre but there is tough resistance from Daesh," said Lieutenant General Abdelwahab al-Saadi, the overall commander of the operation, using an Arab acronym for IS.
After a week of shaping operations aimed at sealing the siege of Fallujah, which lies just 50 kilometres (30 miles) west of Baghdad, elite forces launched a new, more aggressive phase on Monday morning.
But they have so far been unable to reach the city centre and battle IS fighters in the streets.
"Every time our forces try to push in, they encounter really tough defence systems set up by Daesh," said a police colonel, speaking on the outskirts of Fallujah.
The closest Iraqi forces have come to moving into the centre is from the south, where they entered a suburb of Fallujah but were pinned back by a massive counterattack on Tuesday.
Iraqi commanders claim they have killed dozens of IS fighters since the start of the operation on May 22-23 but have been coy about releasing their own casualty figures.
Yet the number of coffins being sent back to some of Iraq's southern provinces and of burials reported in the Shiite Muslim holy city of Najaf suggest that the anti-IS camp is also paying a heavy price.
"Since the start of the operation, we have received about 70 martyrs, probably a bit more," said a member of the security forces posted outside Najaf's Valley of Peace, the world's largest cemetery, where many from Iraq's Shiite majority bury their dead.
Officials in Basra said the southern province had lost 26 fighters from the Hashed al-Shaabi paramilitary force alone.
An official in Najaf province, who did not want his name to be published, confirmed 12 deaths from the province.
Medics also reported many wounded from the battle for Fallujah. Since Monday, just two of the capital's hospitals received 97.
Inside Fallujah, tens of thousands of trapped residents were under increasing pressure from worsening shortages and nervous IS fighters preparing for a desperate holdout.
The United Nations Children's Fund said at least 20,000 of them were thought to be children, most vulnerable to dire living conditions and to forced recruitment as fighters.
"Children who are recruited see their lives and futures jeopardised as they are forced to carry and use arms, fighting in an adult war," the agency's Iraq representative Peter Hawkins said.
No aid has reached Fallujah since September last year and residents have been living on dates, dirty water from the Euphrates and animal feed.
Many Iraqi officers expect IS to put up more of a fight for Fallujah than some of the other cities they have lost in Iraq, such as Tikrit and Ramadi.
Fallujah is one of only two major cities they still control in the country -- the other being Mosul -- and it looms large in modern jihadist mythology.
In November 2004, it took 10,000 highly trained US troops backed by the best in military technology more than six weeks to defeat jihadists in Fallujah, suffering some of their worst battle losses since the Vietnam War.
IS has been on the back foot in Iraq, losing much of the territory it seized in 2014.
But it has also come under growing pressure in the Syrian part of the "caliphate" it proclaimed two years ago.
A US-backed alliance of Kurdish and Arab fighters launched an offensive against IS-held territory along the Turkish border that is seen as a main entry point for foreign fighters.
The advance brought the Syrian Democratic Forces to within 18 kilometres (11 miles) of Manbij, a strategic town held by IS since 2014 which was hit by intensive US-led coalition air strikes, the Syrian Observatory for Human Right said.
The Pentagon has deployed more than 200 special forces alongside the SDF, a Kurdish-led alliance in which it has been trying to boost the Arab element.
"Over the past 24 hours, the SDF have seized control of nine villages... west of the Euphrates," the Observatory said.