south in an apparent effort to shore up Iraqi defences.
Bombs shook the capital as US Army and Marine armoured columns took separate, converging paths toward the city from the south.
"The dagger is clearly pointed" at the heart of Saddam Hussein's regime, said Brig. Gen. Vincent Brooks.
The rapid advances brought thousands of troops within the so-called red zone, an imaginary line on the map near the capital where Iraqi use of weapons of mass destruction is most feared. Troops in some lead Army units donned chemical protection suits, and Marine helicopter pilots were ordered to be prepared to do so.
The Black Hawk helicopter was downed by small-arms fire near the city of Karbala, site of fierce fighting between the US Army's 3rd Infantry Division and Iraqi troops, including Republican Guard forces. Seven soldiers were killed and four wounded and rescued, officials said.
Iraq also shot down a one-seat Navy F/A-18 Hornet with a surface-to-air missile, US Military officials said. There was no immediate word on the fate of the pilot.
Despite the gains on the battlefield, US Defence Secretary Donald H. Rumsfeld cautioned that some of the toughest fighting might lie ahead, seeking to dampen speculation that the war might end quickly.
Iraq insisted the battlefield was tilting its way. "Victory is ours," Saddam said in a statement issued in his name.
A US official said the plan was to begin to form a cordon around Baghdad and press the regime to surrender. Failing that, there was no word on when or whether forces would enter the city of 5 million.
Following days of heavy air strikes, US Army units swept toward Baghdad from the southwest, past battered Iraqi forces near Karbala. The troops moved through a gap west of the city.
Special forces earlier seized control of a nearby dam, erasing fears that Iraqi forces would blow it up and cause a flood that could have been a disaster for the American troops.
Advancing Army forces captured a bridge over the Euphrates River at Mussayib, north of Karbala, after defusing explosives left by retreating Iraqis. Coalition warplanes also dropped nearly 40 bombs on a military storage facility in Baghdad.
Brooks told reporters that the Baghdad Division of the Republican Guard, which was positioned near Kut, "has been destroyed." At the Pentagon, Maj. Gen. Stanley McChrystal said both the Baghdad Division and the Medina Division, near Karbala, were "no longer credible forces."
Officials said remaining Republican Guard units, as well as the Special Republican Guard, have also suffered losses, though not as extensive. Baghdad is also defended by a paramilitary force estimated at between 6,000 and 8,000.
US officials warned of difficult fighting ahead.
"We are not expecting to drive into Baghdad suddenly and seize it," McChrystal said.
While most of the day's military action was south of Baghdad, there was fresh bombing around Mosul, in northern Iraq.
A 40-vehicle convoy carried food, fuel and medicine to American troops from neighbouring Turkey as part of a hard-won agreement with the Turkish government.