Iraqi officials say hundreds of people, including scores of ISIL fighters, have been killed or wounded in the street battles and the army's bombardment of the towns.
It's the worst violence to hit Iraq's Anbar province, which borders northern Syria, since al-Qaida's ferocious battles with U.S. forces in 2004-05.
Latest reports indicate the jihadists and their tribal allies hold virtually all of Fallujah, which was al-Qaida's capital during much of the Iraq war, and most of Ramadi, Anbar's provincial capital.
"Fallujah's under the control of ISIL," a senior government security official in Anbar said Sunday as the army outside the town, supported by local tribal militias who oppose the jihadists' strict interpretation of Islam, shelled it with artillery and mortars.
Iraqi Shiite Prime Minister Nouri al-Maliki visited Washington in October and urged the United States to come to his aid.
The administration of U.S. President Barack Obama has refused to get involved in another Middle Eastern war and rejected his request for Apache AH64A/D helicopter gunships, which could be a deadly weapon against insurgents.
But the Americans agreed to supply Baghdad with Hellfire air-to-ground missiles -- the type used by U.S. drones to attack al-Qaida forces in Yemen and Pakistan -- and intelligence-gathering systems.
It's not clear whether these systems are being used in the Anbar fighting, although there have been reports of some airstrikes against the jihadists.
The province, which covers about one-third of Iraq's territory, is largely controlled by ISIL and is a strategic conduit to its forces fighting in northern Syria where they are struggling to establish a caliphate that would straddle the turbulent border.
This would be based on Baghdad, which has a deep resonance with Islamists as it was the center of the Abbasid Caliphate, the third of the Muslim caliphates to succeed the Prophet Muhammad, who died in AD 632.
The Abbasids took over authority of the Muslim empire from the Umayyads in 750 and ruled until 1258.
Western Iraq has been in ferment for months as al-Qaida, which had been badly crushed before U.S. forces withdrew from Iraq in December 2011, rebuilt its strength, greatly enhanced by the increasingly sectarian nature of Syria's civil war.
The overwhelmingly Sunni region has suffered under a major crackdown by Maliki, who has systematically sought to disenfranchise the Sunni minority who had been the pillar of Saddam Hussein's grotesque regime that was toppled in the U.S.-led invasion of 2003.
That put Iraq's Shiite majority, brutally suppressed by Saddam, in power for the first time. Many old scores were settled.
The wave of suicide bombings and assassinations, directly largely at Shiites and Maliki's increasingly repressive regime, that has swept the country for two years was al-Qaida's revenge.
Anbar erupted last week after Maliki, frustrated with growing Sunni anti-government protests, sent in his army to break up a large protesters' camp in Ramadi.
That turned bloody. ISIL exploited Sunni fury to intensify a campaign of systematically weakening Maliki by attacking his security infrastructure into directly challenging him by seizing Fallujah and Ramadi.
The military's gearing up for a major offensive, and it remains to be seen whether the jihadists will be able to hold on.
ISIL forces in Syria are under sustained attack by nationalist and secular rebels who're as opposed to Islamic rule as they are to Assad's brutality. That's likely to prevent jihadist units crossing into Iraq to support those in Fallujah and Ramadi.
Maliki is making deals with Sunni tribes despite their misgivings about his rule.
Even so, ISIL's expected to go on fighting in both Iraq and Syria.
Yet "for all the dedication and motivation of its fighters, ISIL simply does not have the manpower or the force to overcome its innumerable enemies and achieve its end goals of establishing its version of an Islamic caliphate in Syria and Iraq," the U.S. global security consultancy Stratfor observed.
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