As relentless bombing again shook this city, Iraq said the week-long conflict had caused more than 4,000 civilian casualties, including more than 350 dead. Health Minister Umeed Midhat Mubarak said 36 civilians had died in air raids in the Iraqi capital in the past 24 hours alone.

Iraq said its forces had brought down a US Apache helicopter yesterday, and the Al-Jazeera Arab satellite television channel showed pictures of a downed US helicopter surrounded by Iraqis waving rifles.

A statement on Iraqi state television said forces of the Saddam Fedayeen militia had shot down an Apache and a pilotless US drone, and had destroyed four armored personnel carriers.

Jazeera broadcast footage of a helicopter with US military markings, and of the wreckage of what it said was a US drone.

Bush and Blair stressed they would not be deterred by Iraqi resistance, which has been stronger than expected. They refused to say how long the war might last.

“Saddam Hussein and his hateful regime will be removed from power. Iraq will be disarmed of weapons of mass destruction. And the Iraqi people will be free,” Blair said.

The Washington Post quoted military sources yesterday as saying the war might last months, rather than weeks. Asked about this, Bush said the United States and its allies would fight as long as it took to win.

“This isn’t a matter of timetable, it’s a matter of victory,” the US president said. “And the Iraqi people have got to know that. They got to know that they will be liberated and Saddam Hussein will be removed, no matter how long it takes.”

One independent expert, Col. Christopher Langton, head of defense analysis at London’s International Institute for Strategic Studies, said he thought the fighting would last another month or more.

Frank Umbach, security and defense analyst at the German Council on Foreign Relations, said: “I wouldn’t rule out it dragging on for a couple of months.”

In a battle for world opinion, much of which is hostile to the United States and Britain, Blair accused Iraq of executing British prisoners of war. Brig. Gen. Vincent Brooks, the US Central Command’s deputy operations director for command information, accused Iraq of forcing children to fight by threatening to kill their families.

In Al-Zubayr, 13 miles (20 km) south of the main southern city of Basra, independent reporters said that Iraqi militias, mixing with the local population, were pinning down US and British forces, trapping civilians in the crossfire. In that town alone, residents said as many as 15 civilians had been killed. Tanks, which were expected to roll into Basra early in the war, were still bogged down in battles in civilian neighborhoods with a few Saddam loyalists proving strong enough to hold back the invasion.

The United States and Britain have lost a total of 44 troops killed and 12 missing. US Defense Secretary Donald Rumsfeld said there was no point in other nations trying to mediate a cease-fire.

“I have no idea what some country might propose, but there isn’t going to be a cease-fire,” he said. The prospect of having to take Baghdad house by house and street by street is the biggest nightmare of US military planners. Such an operation could cause high military and civilian casualties.

Rumsfeld was asked about this at a Senate hearing. “It could take some time,” he said.

“Now, they (Republican Guard) are in deployed positions — in revetments — and what will happen is they will get degraded from the air and then attacked by coalition forces.” Richard Myers, chairman of the military Joint Chiefs of Staff, said heavily armed US Apache helicopters and attack jets were striking at the guards, “and at some point — at a time of our choosing — we will engage them.”

As the weather improved in Iraq after two days of blinding sandstorms, US troops that have pushed north from Kuwait toward Baghdad consolidated their supply lines and prepared for major clashes with troops loyal to Saddam.

In the north, 1,000 paratroopers from the 173rd Airborne Brigade took over an airfield after one of the biggest combat parachute drops since World War II.

“This is the beginning of the northern front,” a US defense official said. But military experts said this could take weeks and would require a massive airlift of armor.

Repeated explosions hit central Baghdad throughout the day and into last night.