Karzai: The reason for the NATO and American intervention in Afghanistan was terrorism. Terrorism has not gone away. It has increased.
Lara Logan: When you say that terrorism has increased what do you mean exactly?
Karzai: If terrorism means violence against civilians, if terrorism means violence against our allies. It has increased. It has not abated. It has not gone away.
Karzai says Afghan intelligence reports to him on the presence of foreign fighters - Arabs, Chechens and others, who are captured and killed on Afghan soil.
Karzai: Name them al Qaeda, name them Islamic Movement of Uzbekistan, name them Haqqani, name them Taliban, whatever. They're still there. And they have the ability to continue 10 years on to come and hurt us and kill your troops and kill our troops, kill our civilians. We must then question how come they've returned?
Lara Logan: Well that's a good question. How come?
Karzai: How come they've returned?
Lara Logan: How did that happen?
Karzai: Something must have gone wrong for that to happen.
One place where things have gone wrong is in the mountains of Kunar in the east of the country, which has become al Qaeda's base of operations in Afghanistan today.
These are enemy fighters and leaders filmed there by our Afghan cameraman who visited a number of different enemy camps. We couldn't go ourselves because it's too dangerous for Westerners to travel on their own to Kunar. They told our cameraman that they work side-by-side with al Qaeda and share their ideology.
One Taliban commander agreed to meet with us in the Afghan capital. He's a specialist in suicide bombings trained by al Qaeda. The safest place we could find was the back of a car, and he would only talk if we concealed his identity.
As we made our way through the streets, we had to avoid the city's heavy security and keep our cameras hidden from view.
Lara Logan: Who is behind the "insider attacks", what the Americans call 'insider attacks,' infiltrating the Afghan police and army - is that you?
Taliban: These are Taliban attacks. This is part of our new military strategy. We have our people in the Afghan police and the army. And the orders come from the top.
He told us al Qaeda fighters are rushing to Afghanistan and that he has more than a dozen of them under his command. He also said they have been the driving force that has made the Taliban more lethal on the battlefield.
Lara Logan: Are you the only commander with al Qaeda fighters?
Taliban: There are many groups that have them. We can't do this without them.
Lara Logan: What skills do the al Qaeda fighters bring?
Taliban: They are masters of everything. For example, making IEDs, something we don't know how to do. But they are teaching us. They are also master engineers and good with all weapons. When our weapons break, they are the ones who repair them. We can't do this without them.
While the U.S. has been saying for a long time that al Qaeda in Afghanistan is almost defeated, the U.S. military's own reports from the battlefield reveal a very different picture.
They are rich with detail about al Qaeda's leaders and operations today, confirming the existence of al Qaeda training camps and multiple attack cells. Among those they say they've killed are al Qaeda weapons and explosives experts. In one month, the U.S. says it killed more than 25 al Qaeda leaders and fighters.