The US military commander in Iraq says there are grounds for optimism over the latest security drive.
Gen David Petraeus told the BBC that with two out of the five extra brigades now on the ground in Iraq, there had been fewer sectarian attacks.
He said he would have an idea of the chance of success once all extra troops were deployed in the coming months.
Attacks in Baghdad killed at least four Iraqis on Sunday, while the US announced the deaths of seven troops.
The violence included:
- A car bomb at a popular market in the Shia district of Sadr City, which killed three people and wounded another seven
- Four US soldiers were killed and one wounded when their vehicle was ambushed by a roadside bomb in western Baghdad on Saturday, the US military said
- Also on Saturday, an explosion in the capital's Diyala province killed one US soldier and injured five others and a US marine died while fighting in Anbar province
The new "surge" policy is aimed at curbing sectarian and insurgent violence in Baghdad and Anbar province.
Gen Petraeus said: "By early June, we should then have everyone roughly in place - and that will allow us to establish the density in partnership with Iraqi security forces that you need to really get a good grip on the security situation."
He said there were "encouraging signs", although he added that he did not want to get "overly optimistic at all on the basis of several weeks of a reduced sectarian murder rate".
He said the new operation had led hundreds of families to return to "neighbourhoods that had really emptied out".
But Gen Petraeus also pledged to speak candidly if he thought the operation was not working.
He said: "I have an obligation to the young men and women in uniform out here, that if I think it's not going to happen, to tell them that it's not going to happen, and there needs to be a change.
"In other words, if you can't accomplish your mission, you owe that to your boss - and you owe that, more importantly, to those who are out there serving in the coalition."
Sectarian violence in Iraq rose sharply after an attack on a Shia shrine in Samarra last year.
The monthly Iraqi death toll hit a record high in October, with more than 3,700 people losing their lives, according to a UN report.
In December US President George W Bush announced the deployment of a total of 21,500 extra troops in Iraq.
In early March Defence Secretary Robert Gates approved a request for an extra 2,200 military police to support the security drive in Baghdad.
The Democrats, who now control Congress, have condemned Mr Bush's policy, and some have suggested legislation requiring US troops to return from Iraq if security goals are not met.
Thousands of anti-war protesters have been marching in the US and elsewhere to mark the fourth anniversary this coming Tuesday of the start of the Iraq conflict.
In Washington, the marchers followed the same route as a demonstration in 1967 against the US involvement in Vietnam.