The battle to drive so-called Islamic State (IS) fighters out of the centre of Ramadi is entering its second day as Iraqi special forces fight the militants street by street.
Officials are optimistic that the city will be recaptured by the weekend.
But a spokesman for the US-led coalition has been more cautious, saying a tough fight is in prospect.
Iraqi forces are heading towards the main government complex, and have come up against snipers and suicide bombers.
Ramadi fell to IS in May in an embarrassing defeat for the Iraqi army.
US-led coalition spokesman Col Steve Warren estimates that there are up to 350 IS fighters still in Ramadi in addition to possibly tens of thousands of civilians.
There have been reports that IS has been rounding people up, possibly to use as human shields.
BBC Middle East editor Sebastian Usher says that the offensive in Ramadi appears to be a more effective Iraqi military operation, helped by months of US training.
Notable by their absence, our correspondent says, are powerful Shia militias, who helped recapture Tikrit earlier this year.
Their presence would be too incendiary in the Sunni heartland of Ramadi and the surrounding province of Anbar, he adds.
Security sources told the BBC troops and allied tribesmen, backed by US-led air strikes, had already retaken two districts, and entered two others.
Last month, government forces completed their encirclement of the predominantly Sunni Arab city, about 90km (55 miles) west of Baghdad, cutting off militants inside the centre from strongholds elsewhere in Anbar province and in neighbouring Syria.
Iraqi Counter-Terrorism Service spokesman Sabah al-Numani said its troops, supported by soldiers, police and Sunni tribesmen, began the assault on central Ramadi at dawn on Tuesday.
Sources in the Iraqi military’s Anbar Operations Command told the BBC that engineers had built temporary bridges over the River Euphrates, which flows along the north and west of the city centre.
This had enabled troops to enter directly the al-Haouz district, south-west of the government complex.
By Tuesday afternoon, government forces had retaken the al-Thubat and al-Aramil districts, and entered nearby al-Malaab and Bakir, the sources said.
The Iraqi defence ministry said the jihadists had prevented civilians leaving Ramadi since leaflets warning of an assault were dropped over the city last month.
Sources inside Ramadi told the BBC IS had carried out a campaign of raids and mass arrests of residents in districts still under its control, in an attempt to prevent an uprising in support of the government offensive.
IS has lost control of several key towns in Iraq to government and Kurdish forces since overrunning large swathes of the country’s west and north in June 2014 and proclaiming the creation of a “caliphate” that also extended into neighbouring Syria.
On Monday, analysis by IHS Jane’s suggested that IS had lost 14% of its overall territory in Iraq and Syria, about 12,800 sq km (4,940 sq miles), over the past year.
Despite this, the group has been able to capture new territory of strategic value over the same period, including Ramadi and Palmyra in Syria’s Homs province. It also still controls the Iraqi cities of Falluja, east of Ramadi, and Mosul, in the north. (BBC)