The "basic wheels" of government had come off, Sir John said.
A senior British military officer has told the Iraq inquiry the US would not accept that an organised insurgency was developing in the aftermath of the war.
Sir John Kiszely, the UK's top military representative in Baghdad in 2004-5, said Washington felt a "bunch of no hopers" were behind mounting attacks.
Another officer defended the much criticised US decision to disband the Iraqi army, saying they had no choice.
The Chilcot inquiry is looking into UK policy with Iraq between 2001 and 2009.
In its first month, it has heard from a succession of senior civil servants, diplomats and military commanders about the build-up to the March 2003 invasion, Iraq's military threat and post-war planning.
Former Prime Minister Tony Blair, who is due to appear before the inquiry early in 2010, said on Saturday that he would have supported moves to remove Saddam Hussein by force even if he had known Iraq did not possess weapons of mass destruction.
AT THE INQUIRY
BBC World Affairs correspondent Peter Biles
The reluctance of Washington to admit it was facing a growing insurgency in Iraq in 2004 raises a fundamental question - what would have happened had the coalition tackled the problem differently?
Only by April 2005 was there a US change of view, said Britain's Lt Gen Robin Brims.
There was finally a realisation that it was an insurgency, and needed to be treated as such, "even if the word was frowned upon".
By this stage, Fallujah had become "a safe haven" for the insurgents. With hindsight, there are regrets that this was allowed to happen.
Another talking point today was the length of British military postings to Iraq. For the commanders struggling to find levers of influence in Iraqi society, a six month tour of duty was too short.
Lt Gen Sir John Kiszely said that without care, you could be treated as "passing trade".
His remarks have been criticised by former director of public prosecutions, Sir Ken MacDonald, who said it was proof of an "alarming subterfuge" by the UK government in its support for the war.
Questioned about the situation in Iraq after the invasion, Lt General Sir John Kiszely told the inquiry the situation in Baghdad had deteriorated when he took up his post in October 2004.
Attacks on coalition forces were rising sharply, he said, Iraqi police and army units were severely undermanned, the government machine was not functioning and "little" reconstruction had taken place.
"The rule of law did not really exist in a number of provinces," he said.
He described how US Defence Secretary Donald Rumsfeld initially declined to accept growing signs of an organised uprising against coalition forces, a situation he put down to political pressures.
"Secretary of Defence Rumsfeld had instructed that it was not to be called an insurgency. I think he called it a bunch of no-hopers carrying out some terrorist acts. But it was an incipient insurgency."
This attitude inhibited attempts to deal with the growing violence in the country, he added.
"If you recognise that something is an insurgency then you are using a different method to counter it than if you think it is merely terrorism.
"If you think it is merely terrorism then you use merely counter-terrorist tactics and activities on the basis that if you kill or capture the terrorists then you have probably solved the problem. Which is not the case with an insurgency."
At that time, organised groups were drawing support from people described as "accidental guerrillas" - individuals without a political agenda but resentful of foreigners "invading their space".
November-December: Former top civil servants, spy chiefs, diplomats and military commanders to give evidence
January-February 2010: Tony Blair, Gordon Brown and other politicians expected to appear before the panel
March 2010: Inquiry expected to adjourn ahead of the general election campaign
July-August 2010: Inquiry expected to resume
Report set to be published in late 2010 or early 2011
Although US commanders listened to his concerns, Sir John said these did not initially "permeate" the culture of the US operation in Iraq.
Sir John said it was not until March 2005 that top US commander General Casey issued "ten top tips" for dealing with a counter-insurgency, including treating Iraqi civilians with "respect and decency".
Asked about the UK's own counter-insurgency strategy, a senior British commander said his troops had to "relearn" their tactics because they had been used to conventional warfare.
"We had not, perhaps, expected to be faced with an insurgency," said Lt Gen Jonathon Riley, who commanded a multinational division in the south of the country between November 2004 and August 2005.
"We did have to go through a process, at every level, of reawakening and relearning. And we did it on the job."
New units arrived "much better prepared and with the right mindset" as the nature of the threat posed became clearer, he added.
Lt Gen Riley also defended the decision to break up the Iraqi army after the invasion, which critics say contributed to the escalation in violence, saying the force had effectively "disbanded itself".
"What was left of its infrastructure had been largely torn apart by the population which had lost all respect for its own army, a very bad situation to be in," he said.
The inquiry is expected to publish its findings in late 2010 or early 2011.