(Adds al Qaeda claim paragraph 13)

By Dean Yates and Aseel Kami

BAGHDAD, May 7 (Reuters) - The United States said on Monday its military was expecting to suffer heavier casualties as it pushed into "tougher neighbourhoods" in a bid to crush insurgency in Iraq.

The White House warning came on a day when 25 people were killed near Ramadi in two suicide bombings police blamed on al Qaeda. They were the latest in a string of big car bombings across Iraq in recent weeks that have killed hundreds despite a U.S.-backed security crackdown in Baghdad and outlying areas.

"We are getting to the point now with the Baghdad security plan where there is going to be real engagement in tougher neighborhoods and you're likely to see escalating levels of casualties," White House spokesman Tony Snow said.

"We've known that, been saying it all along. We're getting into some of the grittiest security operations."

Eight U.S. soldiers were killed on Sunday in roadside bomb attacks and were among 12 whose deaths were announced, following an April in which more than 100 died. Since the U.S.-led invasion in 2003, more than 3,300 U.S. troops have been killed.

Monday's first bomb went off in a packed market at Albu-Thiyab, a town northeast of Ramadi, said Tareq al-Thiyabi, a police colonel in Anbar province. Ramadi is the local capital.

He said 13 people were killed at the market, including women and children. Nearly 20 people were wounded.

The second car bomb exploded soon after at a police checkpoint in a town called al-Jazeera, where 12 people including five policemen were killed, he added. More than 25 were wounded.

"They are terrorists. They are from al Qaeda," Thiyabi said, when asked who he thought was behind the two blasts.

RUSSIAN PHOTOGRAPHER KILLED

The town of al-Jazeera is home to many Sunni Arab tribal leaders who formed an alliance against Sunni Islamist al Qaeda last year, opening up a fierce power struggle for Anbar.

The tribal chiefs oppose al Qaeda's campaign of indiscriminate attacks on civilians and the imposition of an austere form of Islam in the areas where the group holds sway in the vast desert region that stretches to Syria.

The eight U.S. soldiers killed on Sunday included six who died along with a freelance Russian photographer in one roadside bomb attack north of Baghdad.

The al Qaeda-led Islamic State in Iraq claimed responsibility for the roadside blast, calling it an attack on "cross-worshipping soldiers". The claim was made in a statement posted on a Web site used by Iraq insurgents.

The Russian ambassador to Iraq named the dead photographer as Dmitry Chebotayev. Chebotayev was in his late 20s.

Iraq's Shi'ite Prime Minister Nuri al-Maliki spoke to U.S. President George W. Bush on Monday in a video conference call, Maliki's office said in a statement. The prime minister noted that stopping car bombs remained a challenge, it said.

Recent big suicide attacks in Anbar, an overwhelmingly Sunni province west of Baghdad, have been blamed on al Qaeda.

Tribal leaders have tried to expel al Qaeda from Anbar, and have had some success in pushing out some of the al Qaeda militants, U.S. military officials have said. (Additional reporting by Mussab Al-Khairalla and Ibon Villelabeitia in Baghdad, Steve Holland in Washington, Inal Ersan in Dubai and Tanya Ustinova in Moscow)

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