Islamic militants controlling a mainly Sunni area west of Baghdad are so well-armed that they could occupy the capital, a top Iraqi official warned Monday, a frank and bleak assessment of the challenge posed in routing the insurgents as a new wave of bombings killed at least 31 people.

Since late December, members of Iraq’s al-Qaida branch — known as the Islamic State of Iraq and the Levant — have taken over parts of Ramadi, the capital of the largely Sunni western province of Anbar. They also control the center of the nearby city of Fallujah, along with other non-al-Qaida militants who also oppose the Shiite-led government.

Iraqi government forces backed by Sunni tribal militias launched an all-out offensive Sunday to seize back control of Ramadi and surrounding areas from the militants. Five government-allied tribesmen were killed in the initial fighting, and 15 tribesmen and government forces were wounded, provincial spokesman Dhari al-Rishawi said.