An overwhelming majority of Americans predict more tough fighting ahead in Iraq, despite the fall of Baghdad, and also worry that winning the peace may prove to be more difficult than winning the war, according to a Washington Post-ABC News poll.

The survey also found that President Bush's overall job approval rating continues to rise in step with the upbeat news from the front. Three in four Americans approve of the job that Bush is doing as president, his best rating since June. Eight in 10 support Bush's decision to go to war, and nearly two-thirds say the war is going "very well" for the United States -- up 19 percentage points in less than a week.

Even as Baghdad residents danced in the streets to celebrate the arrival of U.S. forces in their city, three in four Americans said the war was not over and predicted that troops loyal to Saddam Hussein still posed a serious threat to U.S. and allied forces.

"I'm expecting still more fighting," said Roxie Culp, 44, a homemaker living in Little Rock. "They've made it look easy on TV, and I'm sure it has not been easy for our people over there. I think there is still a lot of danger for our troops, still a lot to be done."

The survey also found that what is emerging as a relatively quick and easy victory may give way to an uneasy peace. Six in 10 said they were at least somewhat concerned that the United States will get bogged down in a long and costly peacekeeping effort in Iraq while one in three disagreed.

"I am very much concerned with what happens next," said Marvin Dismuke, 68, a retired steelworker who lives in West Leechburg, Pa. "We've got a lot of American lives at risk as long as we're in there. It wasn't a bad risk going in there. The people were suffering quite a bit and they are welcoming us now. But we now may be staying a long time, and they will think we were trying to set up a dictatorship ourselves. The rest of the world would look down on us."

Despite those concerns, seven in 10 said the United States should commit troops to remain in Iraq to help maintain the peace -- significantly higher than the proportion who were willing to keep troops in the Middle East after the 1991 Persian Gulf War. But a small majority -- 53 percent -- say the United Nations and not the United States should be in charge of the country until Iraq has a functioning government in place.

"I don't worry that we'll get bogged down," said Carmen McDonald, 36, a homemaker in Morristown, Tenn. "But I do think it will be a long and drawn-out process, and it's going to take a lot of time and effort and work to change their government structure. But it's going to be worth it."

A total of 509 randomly selected adults nationwide were interviewed Wednesday night. The margin of sampling error for the overall results is plus or minus 5 percentage points.

Americans are feeling better about their country than they were earlier this year. Two in three said they were satisfied with the way things were going in the United States, up 29 percentage points since early March before the war began.

Americans also are feeling more positive toward the Iraqi people. Three in four said they believe Iraqis are on the side of the United States -- a view shared by only half two weeks ago.

But the survey also found that four in 10 worried that the war may have damaged relations with allies, such as France, Germany and Russia, which opposed the war. And four in 10 expressed concern that the apparent ease with which Hussein was driven from power might lead the United States to become too reliant on its military force to solve future disputes with other nations.

"I don't think we've helped ourselves" with other countries, said Suzanne Newman, 53, a Web designer who lives in Yonkers, N.Y. "I'm especially concerned that after all this talk of weapons of mass destruction they've gone through most of the country and not found something. Not only are there moral issues to that I think it makes us look very bad. And it certainly doesn't help our relationships with other countries."

Half of the public said the war would not be over until the United States captured or killed Hussein. Many feared that Hussein was a threat until he is safely in allied hands.

"Until they find Saddam Hussein, it's not over and they'll keep fighting," said Ron Butler, 42, an office equipment company service manager who lives in Jacksonville, Fla. "As long as he's out there, he poses a threat to us. He's going to try to come back at us at some point, if he's alive."

Four in 10 said it doesn't matter whether Hussein is alive or dead. What's important, they said, is that he is out of power and out of Iraq.

"At some point he just becomes a person who is an expatriate, who is an unpleasant thing that we have to deal with," said Stephen Matz, an orthopedic surgeon who lives in Baltimore. "If he's alive and the country has been liberated I don't think we should say the war is a failure because he wasn't killed. I think at some point he will become an afterthought."

The survey found that a narrow majority -- 52 percent -- said the war will leave the United States in a stronger position in the world. That is considerably smaller than the 84 percent who expressed a similar view after the end of the 1991 war, which was fought with the blessing of the United Nations.

Nearly two in three -- 64 percent -- said the war will leave the Middle East more stable than it is now.

"In the Middle East the only thing they respect is power," said Craig Winter, an electrical contractor in Massachusetts. "I think we've gained some respect in their eyes, not 'love' and not 'like,' but respect."