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WASHINGTON (CBS News) ― President Bush on Wednesday declared al Qaeda "public enemy No. 1 in Iraq," placing increasing emphasis on the terror network forever associated with the deadliest attack in U.S. history.

The president also seemed to offer another definition of success in Iraq — not a lack of violence, but a livable level for citizens.

In a speech to construction contractors, Bush put a heavy focus on al Qaeda, which carried out the Sept. 11, 2001, attacks that killed nearly 3,000 people. In doing so, he sought more bluntly to cast the unpopular Iraq war in terms that U.S. citizens could connect to their own lives.

"For America, the decision we face in Iraq is not whether we ought to take sides in a civil war, it's whether we stay in the fight against the same international terrorist network that attacked us on 9/11," Bush said. "I strongly believe it's in our national interest to stay in the fight."

On Capitol Hill and across the nation, support for the war has long eroded as sectarian bloodletting gripped Baghdad. In the eyes of Democratic lawmakers and much of the war-weary public, U.S. forces have been dragged into a civil war between Shiites and Sunnis.

"The recent attacks are not the revenge killings that some have called a civil war," Bush told the Associated General Contractors of America. "They are a systematic assault on the entire nation. Al Qaeda is public enemy No. 1 in Iraq."

The Bush administration dates the unleashing of sectarian violence to the 2006 bombing of a Shiite shrine in the town of Samarra north of Baghdad, which triggered reprisal attacks against Sunni mosques and clerics. But recently, Bush said Wednesday, al Qaeda's brutality has failed to provoke the sectarian reprisals that it wants.

U.S. military leaders say sectarian attacks in Baghdad have declined, but overall violence remains high.

Bush warned that "casualties are likely to stay high."

"Either we'll succeed, or we won't succeed," he said. "And the definition of success as I described is sectarian violence down. Success is not no violence. ... But success is a level of violence where the people feel comfortable about living their daily lives."

Before the November election, Bush insisted to the media that the United States was "absolutely winning" the war. In December, he said the United States was neither winning nor losing, then clarified that he meant the U.S. was not succeeding as fast as he wanted.

The White House has repeatedly characterized success as an Iraq that can govern, sustain and defend itself.

The president was not "stepping back" at all from that goal in his speech, Bush spokesman Tony Snow said later.

Asked if describing al Qaeda as the main enemy is more politically persuasive, Snow said: "The characterizations here are not part of a sales pitch. They're an attempt to try to reflect what's going on the ground."

Meanwhile, the U.S. military said Wednesday that the fourth of five brigades being sent to help Iraqi security forces as part of the crackdown had arrived this week.

The influx of troops is an attempt to put a lid on the violence and to buy the Iraqi government more time. But the Iraqi parliament plans to take a two-month summer recess, reports CBS News national security correspondent David Martin. The U.S. ambassador is dead set against that.

"It would be pretty hard to understand … if the council of representatives takes a two-month break when critical issues for the nation's future may still be pending," said Ryan Crocker, U.S. Ambassador to Iraq.

(© 2007 CBS Broadcasting Inc. All Rights Reserved. This material may not be published, broadcast, rewritten, or redistributed. The Associated Press contributed to this report.)