President Bush and Sen. John Kerry (search) rushed back to the campaign trail Friday to try to convince voters they had won their debate over foreign policy and to continue the argument over whether going to war in Iraq had made the nation safer. Three post-debate polls suggested voters who watched the policy-driven confrontation Thursday night were impressed by Kerry, with most of those surveyed saying he did better than Bush.

Kerry's running mate, Sen. John Edwards (search), said Friday he told Kerry after the debate "I think people saw the next commander in chief," and he criticized Bush for failing to acknowledge problems in Iraq. "You can't fix a problem if you're not willing to admit that mistakes have been made and that you have a problem," he told ABC's "Good Morning America."

Bush, however, "felt great" about his performance Thursday night and believed he had articulated the strategy and resolve with which he is fighting the war on terror, White House communications director Dan Bartlett (search) said. "I think he spoke from the heart, spoke with strength about the necessity for our country to fight the terrorists over there so we don't have to face them here at home," Bartlett told ABC. "He had a good time last night."

Sen. John McCain (search), the Arizona Republican who informally advised Bush on how to debate his friend and Senate colleague, told reporters in Miami on Friday that the debate was probably Kerry's "brightest moment" in the last six weeks. "He presented himself well, John did," McCain said. "Kerry came out slugging."

From the first question, Kerry went on the offensive, accusing Bush of leaving U.S. alliances around the world "in shatters" and later calling Iraq "this incredible mess." Bush noted that Kerry voted to authorize the same war he now criticizes. "That's not how a commander in chief acts," Bush charged.

Less than five weeks before the election, Iraq dominated the debate on a day when a string of bombs killed 35 children and wounded scores of others in western Baghdad. Overall, more than 1,000 U.S. soldiers have died in Iraq.

Kerry summed up Bush's strategy for Iraq as "more of the same" and added: "This president has made, I regret to say, a colossal error of judgment. And judgment is what we look for in the president of the United States of America."

Bush acknowledged that not every American agrees with the decisions he's made. "But people know where I stand," Bush said, suggesting they don't know where Kerry stands. "People out there listening know what I believe."

From Florida, Bush was heading out Friday to rallies in Pennsylvania and New Hampshire, two key battleground states. Kerry was spending the day campaigning in Florida, where the presidential race was decided four years ago.

In Thursday night's encounter at the University of Miami, Bush and Kerry drew heavily on oft-repeated lines from their campaign speeches but they faced each other directly across the same stage for the first time.

Bush appeared irritated when Kerry leveled some of his charges, scowling at times and looking away in apparent disgust at others. Kerry often took notes when the president spoke. The television networks offered a split screen to viewers so they could see both men at the same time and watch their reactions.

On Iraq, Bush criticized Kerry for saying it was the wrong war at the wrong time in the wrong place. "What message does that send to our troops?" the president said. "Not a message a commander in chief gives."

Repeating a line he has used countless times to show his opponent is inconsistent, Bush tweaked Kerry for saying he voted for an $87 billion spending bill for Iraq and Afghanistan before he voted against it.

Kerry shot back, "Well, you know, when I talked bout the $87 billion, I made a mistake in how I talk about the war. But the president made a mistake in invading Iraq. Which is worse?"

Trying to persuade voters that he is tough enough to be commander in chief, Kerry said, "I believe in being strong and resolute and determined. And I will hunt down and kill the terrorists, wherever they are." He said that Bush, in invading Iraq, lost sight of the goal of capturing terrorist leader Usama bin Laden.

But Bush insisted that "the world is safer without Saddam Hussein." He called Iraq "a central part in the war on terror" and said 75 percent of bin Laden's leadership had been brought to justice.

Trying to turn Kerry's criticism against him, Bush said, "I understand what it means to be the commander in chief. And if I were to ever say, 'This is the wrong war at the wrong time at the wrong place,' the troops would wonder, 'How can I follow this guy?'"

Kerry said Bush failed to exhaust diplomatic solutions before ordering the invasion of Iraq and that the United States was left with 90 percent of the cost and 90 percent of the casualties. "Now we have this incredible mess in Iraq — $200 billion," Kerry charged. "It's not what the American people thought they were getting when they voted."

Bush ridiculed his opponent, saying he denigrated U.S. allies in the war, voted against the $87 billion measure to aid Afghanistan and Iraq and sent mixed signals. To Kerry's contention that he could summon broader international support for the war, Bush said, "They're not going to follow someone whose core convictions keep changing because of politics."

While Iraq was the dominant issue in the debate, there were notable differences on North Korea and Iran, two nations suspected of pursuing nuclear weapons programs. Kerry urged that the United States hold direct bilateral talks with North Korea, but Bush called Kerry's proposal "a big mistake" that would crush multinational talks and remove pressure from China on North Korea. Kerry said North Korea has amassed more nuclear weapons during Bush's administration.

On Iran, Kerry said the United States should have worked with allies like France, Germany and Britain to impose sanctions if Tehran refused to give up its nuclear program.