Heavy fighting in the Baghdad slum of Sadr City between insurgents loyal to Shiite cleric Muqtada al-Sadr (search) and U.S. forces killed as many as 42 people Tuesday, American and Iraqi officials said.

One U.S. soldier was among the dead, and up to 203 people were wounded. Five other American soldiers were killed in separate attacks in and around Baghdad on Monday and Tuesday and three Defense Department contractors also died, bringing the U.S. death toll since the war began to 1,003. That figure includes those military personnel killed in combat — 754 — and in non-hostile incidents — 246 along with the three contractors, according to the Pentagon.

"We certainly honor the courage and sacrifice of every man and woman in uniform who has served in Iraq and who is currently serving there. And needless to say we mourn with the families of those lost," Defense Secretary Donald Rumsfeld (search) told reporters at the Pentagon on Tuesday.

The U.S. military has not reported overall Iraqi deaths. The Iraqi Health Ministry started counting the dead only in April when heavy fighting broke out in Fallujah and Najaf. However, conservative estimates by private groups place the Iraqi toll at at least 10,000 -- or 10 times the number of U.S. military deaths.

In the violence in Sadr City, tanks, armored personnel carriers and Bradley fighting vehicles roamed the streets in the sprawling neighborhood of 2 million Shiites as warplanes flew overhead. Ambulances wailed on their way to hospitals as smoke filled the sky.

Tuesday's violence was the latest instance of a persistent insurgency effort within Iraq to thwart the U.S.-led coalition in its efforts.

Joint Chiefs Chairman Gen. Richard Myers conceded Tuesday that there had been a recent spike in casualties — among Iraqis, U.S. forces and insurgents — in Iraq.

"The enemy has become more sophisticated in efforts to destabilize the country," Myers said during a press conference at the Pentagon. "This is a pattern across Iraq." But, he added, the United States would continue to pursue the insurgency.

Nevertheless, both Myers and Defense Secretary Donald Rumsfeld expressed hope that brighter days were ahead for Iraq and also for Afghanistan, where American military operations continue.

"As we move closer to elections in those countries, the terrorists and former regime elements are going to recognize how close they are to losing their opportunities," Rumsfeld said. "The question is, are the people going to win back their country from the extremists? ... I think in both cases we'll be successful."

Sadr City (search) was called "Saddam City" until the U.S.-led invasion, when it was renamed for al-Sadr's slain father, a revered ayatollah killed by Saddam Hussein's agents in 1999.

In another part of the capital, a roadside bomb targeted the Baghdad governor's convoy, killing two people but leaving him uninjured, the Interior Ministry said. Three of Gov. Ali al-Haidri's bodyguards were also hurt.

In addition, two Italian women, an Iraqi woman and an Iraqi man were abducted Tuesday outside the Baghdad offices of the Italian aid agency they all worked for called "A Bridge To...", the agency said. Another Iraqi man managed to escape. The Italian agency is involved in water projects and school construction.

Simona Pari and Simona Torretta, both 29, work for an NGO called "A Bridge To ...," the press office at the agency's Rome headquarters said. The two Iraqis were identified as Raad Ali Aziz and Mahnaz Bassam.

A spokesman for the organization, Lello Rienzi, told reporters in Rome that about 20 armed men stormed their offices, saying they were from an unidentified "Islamic group."

"We had no sign of danger," Rienzi said.

Also Tuesday, a day after a deadly homicide bombing in Fallujah that killed seven American Marines and three Iraqi soldiers, U.S. warplanes launched airstrikes at the city.

Tanks fired into the Sunni insurgent stronghold of Fallujah late Monday night, the U.S. military said.

At least nine Iraqis were wounded in the strikes, said Adel Khamees of the Fallujah General Hospital.

Marine spokesman Lt. Col. T.V. Johnson said insurgents attacked American positions outside the city and U.S. forces retaliated with strikes on suspected militant hideouts inside Fallujah. He said warplanes fired several missiles, and artillery units and tanks were also in action.

He said there were no reports of American casualties.

Labor Day was the deadliest day for American forces in four months. A group linked to Jordanian-born militant Abu Musab al-Zarqawi (search) — Tawhid and Jihad — posted a statement on a Web site Tuesday claiming responsibility for the fatal bombings.

That attack underscored the challenges U.S. commanders face in securing Fallujah and surrounding Anbar province, the heartland of a Sunni Muslim insurgency bent on driving coalition forces from the country.

Rumsfeld said the terrorists and insurgents in Iraq had underestimated "our country, our coalition." The defense secretary then addressed the criticism that it was the other way around and the United States had underestimated the insurgents.

"There have been critics of every war," he told reporters. "Certainly it's understandable."

But he also defended the operation, saying those involved were making slow but steady progress and the country was moving toward establishing its own democracy.

The fighting in Sadr City erupted when militants attacked U.S. forces carrying out routine patrols, killing one American, said U.S. Army Capt. Brian O'Malley.

A senior Health Ministry official, Saad al-Amili, said a total of 35 people have been killed and 203 injured in the Sadr City clashes. One of the dead was an American soldier killed in a rocket-propelled grenade strike.

A Health Ministry spokesman told FOX News 41 Iraqis had been killed and 170 wounded.

An al-Sadr spokesman in Baghdad, Sheik Raed al-Kadhimi, blamed what he called intrusive American incursions into Sadr City and attempts to arrest the cleric's followers.

"Our fighters have no choice but to return fire and to face the U.S. forces and helicopters pounding our houses," al-Kadhimi said in a statement.

In the slum's roadways, small groups of Sadr's Mahdi militia (search) fighters used hammers to dig up the asphalt to plant explosives. Bands of fighters in civilian clothes — mostly in their teens and early 20s — wielded rocket-propelled grenades and trotted toward the clashes, children running in their wake.

Other fighters, rifles in hand, gathered on street corners. Roads leading to the area were blocked by the militiamen using rocks and tires. By early afternoon, most stores in the neighborhood were shut in anticipation of more combat.

The renewed fighting came after a period of calm in the impoverished neighborhood after al-Sadr called on his followers last week to observe a cease-fire and announced he was going into politics.

But al-Sadr aides later said peace talks in Sadr City between the cleric's representatives and interim Prime Minister Ayad Allawi's government had stalled, with the government refusing militants' demands for U.S. troops to keep out of the troubled district.

U.S. commanders have said they want to carry out an assault to clear al-Sadr's fighters from the district, particularly its northern part where the militiamen are said to have dug in, setting explosives and booby traps.

Al-Sadr led a three-week uprising in the holy city of Najaf (search) that ended 10 days ago with a peace deal that allowed his Mahdi militia fighters to walk away with their guns. The combat in Najaf left thousands dead and devastated much of the city.

But many Mahdi militiamen are believed to have returned to their stronghold in Sadr City.

U.S. forces have not patrolled in Fallujah since a three-week siege of the city in April that was aimed at rooting out militiaman. As a result, insurgents have strengthened their hold on the city, using it as a base to make car bombs and launch attacks on U.S. and Iraqi government forces.

The Defense Department's most recent published count, as of Friday, shows 976 U.S. service members dead.

In other violence in Iraq:

— The son of the governor of the northern city of Mosul was killed in a drive-by shooting Tuesday, hospital officials said.

— Unknown gunmen killed the deputy director of Baghdad's al-Karama hospital, the Health Ministry said. The motive for the attack was not known.

— Two Iraqi policemen were killed and two others injured in a drive-by shooting in Latifiyah, 25 miles south of Baghdad late Monday, police said.

FOX News' Catherine Donaldson-Evans, Orlando Salinas and The Associated Press contributed to this report.