To paraphrase the great newsman Paul Harvey, we usually think we know what the news is, but it's often difficult to get "the rest of the story." That's especially true when it comes to getting accurate information out of Iraq.

But some news is getting out. And plenty of it is good.

Rep. Steve King recently returned from his fifth trip to Iraq. The Iowa Republican told me of a vast improvement in troop morale since the troop surge started a few months ago.

King says our troops are successfully capturing territory and pacifying towns, and impressing Iraqis in many ways. One of our warriors told King the Iraqis even suspect our forces wear air-conditioned uniforms -- how else could they work so well in the stifling heat?

Some Democrats are also impressed. Sens. Richard Durbin, Jack Reed and Bob Casey -- all frequent critics of the Bush administration -- have commented recently that the U.S. is making military progress in Iraq.

Lawmakers aren't the only people seeing progress. On July 30 The New York Times ran an op-ed piece that called Iraq "A War We Just Might Win" by scholars Michael O'Hanlon and Kenneth Pollack of the left-leaning Brookings Institution. Both men have frequently criticized the Bush administration's approach to the Iraq war. But they're now impressed with our military progress.

Furthermore, they write, we're targeting our most dangerous international enemy: Islamic extremists. "In war, sometimes it's important to pick the right adversary, and in Iraq we seem to have done so. A major factor in the sudden change in American fortunes has been the outpouring of popular animus against al Qaeda and other Salafist groups, as well as (to a lesser extent) against Moktada al-Sadr's Mahdi Army," they wrote.

It's a critical point, because Iraq is a major front in the greater war against terrorism. As Gen. David Petraeus explained recently, al Qaeda in Iraq is very much a part of Osama bin Laden's terrorist organization.

"[It's] very clearly linked to the so-called AQSL, the al Qaeda senior leadership, located in the Pakistan-Afghanistan border tribal areas, without question," Petraeus told radio talk show host Alan Colmes. Thus, the war in Iraq has allowed our military to weaken al Qaeda.

"We have been able to damage very seriously the media operations of al Qaeda in Iraq and their communications ability. In fact, we even killed the three al Turki brothers, who were former[ly] in the Afghanistan area [and] were sent over to al Qaeda in Iraq to help shore up the situation in northern Iraq, which has been under particular pressure in the last several months," Petraeus said.

Our enemies know the stakes, and they're aiming for nothing less than victory. For example, just a few months ago Adam Gadhan, the American spokesman for al Qaeda who's known as "Azzam the American," announced that, "A pullout from Iraq alone, in the absence of compliance with the remainder of our legitimate demands, will get you nowhere, and will not save you from our strikes. So stop wasting your time and trying to save face with these futile farcical maneuvers on Capitol Hill and start making some serious moves."

He seems to have been referring to the congressional debates over whether to set a timetable for withdrawal from Iraq. But (whether he knows it or not) he's making a more important point: We can choose to leave Iraq at any time. But, unless we've defeated our enemies there, they (and other Islamic extremists around the world) will celebrate our pullout as a great victory, and they'll increase their efforts to attack Americans, here in the U.S. and overseas.

His cohort Abu Hamza al-Muhajir, leader of al Qaeda in Iraq, made that very point last October: "We will not rest from our Jihad until we are under the olive trees of Rumieh and we have blown up the filthiest house -- which is called the White House." The U.S. simply can't afford to ease up until we've defeated the men with this attitude.

We're finally making progress in Iraq -- and progress in the greater war against Islamic extremists. So, even though Americans are usually impatient when it comes to wars, let's celebrate our recent gains, and give our men and women in uniform what they want: the chance to finish the job in Iraq and come home as victors.

Ed Feulner is president of the Heritage Foundation