“I’m just not sure there’s a connection between terrorism and the war in Iraq,” said Ann Davis, a Republican homemaker in Lima, Ohio, interviewed in a follow-up to the initial survey. She said she was fully supportive of American troops in Iraq but, “I feel we should not be over there, they should be able to figure it out on their own.”

However, another Republican, Marty Woll, 56, a retired accountant from Los Angeles, said he saw a clear link between the war and attempts to combat terrorism. “Iraq was obviously not the precipitating location for the 9/11 attacks, but if you look at the Middle East as a whole, you see it has been spawning the most violent and the most desperate of the attacks,’’ Mr. Woll said. “Saddam Hussein killed almost a million of his own people. That magnitude indicated that someone had to do something about it.”

Mr. Bush’s inability to improve his overall standing despite gains on the terror issue could be traced to people like Lucia Figueroa, 23, an independent from Fort Drum, N.Y., who backs the president on terrorism but faults him elsewhere. “Even though I approve of the way Bush is handling terrorism, he isn’t putting enough focus on other issues, like health care and Katrina, and those things need more attention,” she said

As recently as Monday, Mr. Bush, at a news conference, defended the invasion of Iraq as essential to preventing more domestic terror attacks and said he expected troops to stay there through the remainder of his presidency.

“If you believe that the job of the federal government is to secure this country, it’s really important for you to understand that success in Iraq is part of securing the country,” said Mr. Bush.

But Democrats in recent weeks have made a concerted effort to portray the war in Iraq as a distraction from essential antiterror initiatives, and the poll indicates that message may be effective. Democrats contend that the war in Iraq has sapped resources and attention from tracking terrorists and bolstering domestic security. “We took our eye off the real war, the war on terror,” said Senator Harry Reid of Nevada, the Democratic leader, in a conference call with reporters today.

The public’s judgment on the job that Congress is doing remains largely negative, with 60 percent registering disapproval. Forty-seven percent of the registered voters surveyed in the poll said they expected to vote for a Democrat for the House this November; 32 percent said they would vote for a Republican. The national poll cannot measure the races in individual Congressional districts, but the findings are indicative of the two parties’ relative strength.