MOSUL, Iraq — With anti-Islamic State group forces on the offensive in both the Iraqi city of Mosul and the Syrian city of Raqqa, Iraq’s prime minister on Thursday declared an end to the extremist group’s self-proclaimed caliphate.

But even as Prime Minister Haider al-Abadi made the bold assertion, deadly fighting continued in Mosul — filling field hospitals and forcing hundreds to flee.

“We are seeing the end of the fake Daesh state. The liberation of Mosul proves that,” al-Abadi said on Twitter, using the Arabic acronym for IS. “We will not relent. Our brave forces will bring victory.”

Across the border in in Raqqa, coalition officials predicted a long, bloody battle ahead for the Kurdish-led Syrian Democratic Forces, whose fighters succeeded in completely encircling the militants’ de-facto capital Thursday. U.S.-led coalition officials estimated that as many as 2,500 IS fighters remained in the city.

Beginning at dawn, Iraqi forces began a push deeper into Mosul’s Old City, where IS fighters were making their last stand. The Iraqi troops moved slowly along foot paths strewn with rubble, twisted metal and downed power lines.

Many front-line positions were only reachable by climbing in and out of homes, across roof tops and through holes blasted into concrete walls.

By early afternoon they had reached al-Nuri Mosque, at once a hugely symbolic win and a ruined prize. The site is where IS leader Abu Bakr al-Baghdadi made his only public appearance in July 2014, declaring the self-styled Islamic “caliphate” encompassing territories then-held by the extremists in Syria and Iraq.