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Mr Bush plans to send 21,500 additional US troops to Iraq
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No-one is declaring victory, but the first few days of the new security operation in Baghdad have seen a significant fall in violence.
The commander of the US side of the operation, Maj Gen Joseph Fil, said: "There's an air of suspense throughout the city... Many of these extremists are laying low and watching us to see what we do and how we do it."
The Americans are working to a much more aggressive military doctrine than before.
The frequent presence of a B1 "Lancer" bomber - making a huge din as it flies low over the skies of the city day and night - is a reminder of that.
This is the first time since the invasion in 2003 that fixed-wing bombers of this type have been seen here routinely.
US President George W Bush described the change simply at a news conference earlier this week, saying that up to now America has been good at "clearing ground" but not "holding" it.
Three stages
Gen Fil has now defined much more precisely what the new doctrine means on the ground.
The operation will come in three phases.
It is difficult to see how America can keep up its commitments here at this level for very long
The first will be to clear the ground - moving in and removing any obvious threats.
But the aggression comes in the second phase, controlling the ground, and the pace of the operation will depend on the ability of troops to do that, so they can move onto phase three - retaining it.
There are 10 districts in Baghdad, and the Americans are planning to quickly establish a major forward base in each of them, working alongside Iraqis.
Smaller neighbourhood operating bases will be set up as security allows.
These though, will be tempting targets for insurgents armed with mortars and rockets, as Britain found when they tried a similar tactic in Helmand province in southern Afghanistan last year.
In Baghdad, the more assertive roadblocks have slowed traffic to a crawl, making them an easy hit for bombers.
On Thursday, the day Prime Minister Nouri Maliki said the operation had officially begun, a bomber in the southern district of Doura simply left his car in the traffic chaos near a checkpoint and walked away. Four people were killed when he detonated the bomb, and more than a dozen injured.
The main market in the centre of Baghdad is now a pedestrian-only zone, to stop car bombs.
Gen Fil acknowledged that the insurgents will find other targets.
He said the insurgents had "a thirst for blood like I have never seen before".
'Achilles heel'
Rebuilding infrastructure, and improving job prospects in a city where unemployment is incalculably high, will go hand in hand with the military operation, the US has said.
And in this early assessment of how things were going it was significant that Gen Fil said he still needed more engineers to come in alongside the brigade-strength of US infantry now pouring into Baghdad every month.
Success depends on securing better power and water supplies, as well as other improvements to the well-being of the people of Baghdad.
Gen Fil also said he had asked for more combat helicopters and this could prove to be the Achilles heel of the operation.
The US has lost five helicopters to insurgent fire in a month, and there are new weapons out there that are doing the damage.
The modern history of guerrilla warfare has harsh lessons for Iraq.
The former Soviet Union lost the war in Afghanistan in 1985, six years after its invasion, only when the US supplied lightweight "stinger" anti-aircraft missiles to the mujahideen.
The will of the US was sapped in Vietnam when they began to lose helicopters on a huge scale.
There is another major weakness in the plan too, and that is the dependence on Iraqi troops.
Gen Fil said their quality was growing every day. But privately, US and British forces who have trained the troops are concerned about the Iraqis' ability to do the job.
There was a mutiny of Iraqi troops in the south when they were ordered to go to Baghdad.
British hopes to hand over another southern province at the end of last year have been postponed because of the incapacity of the Iraqi forces there, and the stated aim of Foreign Secretary Margaret Beckett to hand over Basra "by the spring" is not conceivable on present trends.
'Overstretch'
It is difficult to see how the US can keep up its commitments here at this level for very long.
The outgoing US Army Chief of Staff, Gen Peter Schoomaker, has told a congressional committee that the demands placed meant that the army now has to go to "extraordinary lengths to respond".
The five new brigades promised for the security of Baghdad will require translators and support staff, as well as huge amounts of equipment.
The committee heard that 40% of US equipment is in Iraq or Afghanistan or in maintenance, reducing readiness for other conflicts.
As well as the pressure on kit, Gen Shoomaker talked about the impact on the soldiers themselves, faced by repeated tours to Iraq and Afghanistan.
Concern over "overstretch" has also been voiced increasingly in public by senior British officers.
The calculation of the insurgents, already reducing their attacks, could be to wait out the summer, hoping the US does not have a long-term commitment.
The calculation of their opponents is that they can turn Baghdad round, and with it Iraq, by soaking the streets with well-armed and well-trained soldiers and police.
One Iraqi military unit has been patrolling central Baghdad with a megaphone broadcasting this message: "We will hit with an iron hand anyone who takes us on."
Their success will be measured by what follows the iron hand.
