FALLUJA, Iraq (CNN) -- As U.S. soldiers advanced into southern Falluja on Friday, violence and combat intensified across Iraq with battles flaring in Mosul, Baquba and Baghdad.

Adding firepower to an offensive that began Thursday, U.S. airstrikes hit a cemetery in southwestern Mosul on Friday.

The targets were insurgents who carried out attacks on government facilities and Iraqi forces in the northern Iraqi city earlier in the week, said Capt. Angela Bowman of Task Force Olympia.

Iraqi National Guard forces and soldiers from the 1st Brigade, 25th Infantry Division, or Stryker Brigade Combat Team launched the offensive Thursday at the request of the governor of Nineveh province, where Mosul is located.

A U.S. soldier was killed by small arms fire during the offensive, a military statement said. The soldier was assigned to Task Force Olympia. No other details were immediately available.

Clashes also erupted in al-Bakr neighborhood, a Kurdish neighborhood, between Patriotic Union of Kurdistan members and insurgents, who tried to take over a PUK building.

The violence across Iraq comes as the imam of the Abu Hanifa mosque in al-Adamiyah called for a jihad against U.S. forces and the interim government during Friday's noon prayers.

In Baquba, a Sunni Triangle city northeast of Baghdad, insurgents and Iraqi police forces battled early Friday afternoon near the al-Sharif cemetery.

Two Iraqi police and two Iraqi civilians, both of whom are women, were wounded, a hospital official said. An official from the Baquba hospital adds that both women are in critical condition. Baquba is in Diyala province. Attacks in Baghdad

In Baghdad, one soldier died and three other people were wounded when a Task Force Baghdad patrol was ambushed Friday afternoon in the southern section of the city.

Northeast of the capital, insurgents shot down a UH-60 Blackhawk helicopter Friday afternoon, the U.S. military said.

"Four multinational forces were on board the aircraft at the time. Three of them were wounded, but injuries are unknown at this time," the military said in a statement.

And the al-Adamiyah neighborhood in the northern part of Baghdad saw insurgents attack an Iraqi police station and Iraqi police patrols on Friday, according to Iraqi police.

The insurgents used small arms fire and rocket-propelled grenades. There were no reports of casualties. Troops push into southern Falluja

U.S. troops fought small cells of insurgents in alleyways and bombed-out buildings as the all-out assault on the Sunni Triangle flashpoint city of Falluja entered its fifth day.

CNN's Jane Arraf, who is embedded with the Army, reported Friday the soldiers are clearing the way for Marines, who are going door-to-door in an effort to find weapons caches and secure the area.

"They have essentially taken the south," Arraf said.

U.S. forces hope to take over the last rebel bastion in southern Falluja during the night, a U.S. Marine officer told Reuters news agency on Friday.

Tank company Capt. Robert Bodisch also told Reuters dozens of insurgents had been killed or captured in the south Falluja stronghold.

Eighteen U.S. troops and five Iraqi soldiers have died in the Falluja operation, Marine commander Maj. Gen. Richard Natonski said Thursday. In addition, the Combined Press Information Center said 178 U.S. service members and 34 Iraqis have been wounded in the operation.

Dozens of wounded troops are being airlifted to Landstuhl Regional Medical Center in Germany.

The army unit cleared an industrial section of Falluja "where they found at almost every turn buildings wired to explode, bombs in the making, anti-tank mines, weapons lying around," Arraf reported. "The whole place was an arms cache."

The U.S. military has pounded the industrial area with airstrikes, wire-guided missiles and artillery fire to wipe out most large bands of insurgents.

"There isn't a block where there hasn't been a building that has been flattened, a tank round through it or a bomb dropped on it," Arraf reported.

Small cells of three to five men continue to fire at the soldiers and hide in the urban area.

Fighting raged through the night, with light flickering in the night sky amid artillery and tank fire.

An Iraqi Intervention Forces company commander, working with the U.S. Army's 1st Infantry Division, said Friday the mission is meeting less resistance than expected, noting foreign fighters have been found among the bodies of insurgents killed in the fighting.

A spokesman for Iraq's interim prime minister Friday said a number of foreign fighters were detained in Falluja fighting.

They include 10 from Iran, one from Egypt, one from Sudan, one from Saudi Arabia and one from Jordan, according to Thair al Nakib.

As of Thursday, the Pentagon said, more than 500 insurgents have been killed in the Falluja offensive.

Earlier, U.S. and Iraqi forces seized the mayor's office, other government offices, and several mosques and bridges in their push to retake the city from insurgents, military officials said.

Operation New Dawn -- intended to pacify the city ahead of the scheduled January elections for a transitional national assembly -- got going Sunday night with the seizure of a hospital and the securing of two bridges over the Euphrates River. (Gallery)

But the actual offensive began in earnest Monday when 10,000 U.S. soldiers and Marines, aided by 2,000 troops from Iraq's new army, stormed Falluja.

Falluja was considered an insurgent command-and-control center for the rest of the country and a base for Abu Musab al-Zarqawi's terror network.