Maj. Gen Stanley McChrystal said coalition troops had captured key bridges over the Euphrates River and were trying to isolate irregular forces in the towns of As Samawa and An Nasiriyah.
The air sorties targeted the Medina, Hammurabi, Baghdad and Al Nida division, McChrystal said, adding: "We are seeing significant degradation of those forces."
McChrystal said since the operation began, the coalition had launched 700 Tomahawk Cruise missiles and dropped 8,000 precision-guided missiles on Iraqi targets, 3,000 of them in the past three and a half days.
Pentagon spokeswoman Victoria Clarke said there was evidence that Saddam Hussein's family and family members of high-level officials were trying to flee the country. Asked whether that came from intelligence our anecdotal reports she declined to elaborate.
"We have seen some reports lately, and I'll just leave it at that," she said.
Coalition officials said Central Command chief Gen. Tommy Franks has decided to go for Baghdad without the full reinforcements of the 4th Infantry Division, to beat the arrival of the intense heat of a desert summer.
His decision came after intense discussions over the weekend among coalition command staff and with Washington, officials said. Pentagon officials say they have been adding about 2,000 troops a day to the ground forces.
McChrystal said the plan all along had been to phase in troop emplacements, so they could halt deployments if they found they were not needed.
Despite widespread reports of an "operational pause," coalition forces advanced Monday beyond the city of Hilla, within 30 miles of Baghdad's outskirts, as the Republican Guard's Medina division began to crumble under the force of air and artillery strikes.
Forced out of their bunkers and forced to give away their concealed firing positions to tackle ground probes from the 2nd brigade of the U.S. 3rd Infantry Division, the Republican Guard fell prey to prowling U.S. fighter bombers and combat helicopters.
McChrystal said they were looking for the "tipping point," at which the Iraqi forces were in such disarray they couldn't effectively fight.
He said there were signs that point is being reached in Basra, in southern Iraq, and in An Nasiriyah.
He acknowledged that coalition forces hadn't yet found biological or chemical weapons but were targeting suspected storage areas, as well as facilities for delivering those agents.
"We still believe strongly that the regime has the capability and potentially the intent to employ those weapons," McChrystal said.
U.S. President George W. Bush was in Philadelphia on Monday talking to the Coast Guard about homeland security. The president said that the Coast Guard would be one of the main lines of defense for U.S. shores if Iraq decided to send terrorists to the United States.
He said Saddam "may try to bring terror to our shores."
He said though "our enemies are desperate (and) we know that they're dangerous," the U.S.-led forces wouldn't not stop until Iraq was free of Saddam.
Clarke repeated the same message. Questioned whether it was premature to believe there was a feeling among the Iraqis that the outcome of a U.S. victory is certain, she said: "It's not a feeling that we will achieve, it is the inevitable outcome that the Iraqi regime will be ended, the Iraqi people will be free of decades and decades and decades of torture and oppression."
After a three-day halt for resupply, the 5th Marines began moving north again toward an undisclosed location. Three of the Marines were injured Monday while digging defense positions when a cluster bomb suddenly exploded.
In the town of southern town of Najaf, where members of the U.S. Army's 101st Airborne Division seized an airfield, troops from the 3rd Infantry Division fired on a van carrying women and children when it did not stop. Seven people were killed and four wounded, Centcom said.
Warning shots were fired at the approaching vehicle, but these were ignored, a CENTCOM statement said. Shots at the vehicle's engine were also ignored.
"It continued moving toward the checkpoint," the statement said.
"As a last resort, the soldiers fired into the passenger compartment of the vehicle," the statement said.
The town is the scene of last week's suicide bombing that killed four U.S. Marines. The Iraqis promised more such attacks and said they had more than 4,000 volunteers from the Arab world who wanted to engage in suicide operations.
U.S. officials, however, dismissed the move.
"It's not a very effective military tactic at all," General Vincent Brooks said in Qatar, where Centcom's forward base is located. "It's a terror tactic and it won't be effective."
Iraqi officials said Monday they have killed hundreds of coalition forces, saying that 43 had been killed in 24 hours. Department of Defense officials have acknowledged 43 U.S. deaths and 29 among British forces since the operation began. There are 109 wounded, seven U.S. forces taken prisoner, and 17 unaccounted for.
Also Monday, the first two trucks since the war began carrying U.N. relief supplies arrived in northern Iraq under the oil for food program, a U.N. spokesman said. There is $14 million in aid relief supplies on the way and a 40-truck convoy will enter soon.
In Geneva, Switzerland, Ross Mountain, chairman of the U.N. task force on Iraq, told reporters Monday the world body was considering ways of getting humanitarian aid to Baghdad's 5.5 million inhabitants on the eve of the battle over the city. He said trying to secure humanitarian corridors into the southern city of Basra and other parts of the war-torn country was among the issues under urgent discussion at the U.N. headquarters in New York.
(Reported by Martin Walker in Kuwait City; Claude Salhani in London; Nicholas Horrock in Ankara, Turkey; Ghassan al-Kadi in Baghdad; and John Zarocostas in Geneva)