"I don't take it seriously at all," Karzai said. "I'm more concerned about the loss of life yesterday in Kabul."
Twenty-six people were killed in an explosion in the capital's downtown area.
Shortly after the attack on his life Thursday, he had said he expected such attempts.
Still, he promised to be more careful about his personal security. "I will not be as reckless as I am."
At the Pentagon Friday, Air Force Brigadier General John Rosa said Karzai's security detail, which included U.S. Special Forces troops, performed as expected during the shootout.
"They did exactly what they were supposed to do," he said at a Pentagon briefing. "After shots were fired, they fired back.
"There's no indication that they did anything except what they were supposed to do, and do it very well," he added.
Still, following the attempt in southern Afghanistan, security was tightened around Karzai, reports CBS News Correspondent Lara Logan.
Meanwhile, the investigation into the car-bombing in downtown Kabul Thursday that claimed 26 lives continued.
U.S. Defense Department spokesman said the blast was intended to destabilize the new Afghan government. Torie Clarke added that the U.S. military suspects that remnants of the Taliban may be responsible.
Karzai, however, said the attacks were the acts of individual terrorists, not a threat to his government.
"These incidents do not indicate any problems," Karzai told reporters.
"These are incidents done by terrorists in an isolated manner," he said. "This means they are no longer capable of mobilizing as groups so they act as individuals."
Nevertheless, Karzai said Afghanistan must improve its security.
"We have a long way to go to bring total technical security to the country," he said. "So it is a job that we have to do in cooperation with the rest of the world."
Logan reports Afghan authorities have confirmed the arrest of one local man and possibly two in connection with the car bomb attack in Kabul, but a foreign ministry spokesman promised there will be more. He said they will use every lead they have and pick up anyone even remotely connected to this investigation, a sign of how seriously the interim government is taking these attacks.
The man who fired his rifle into Kazai's car, missing the president by just inches and grazing the Kandahar governor, had been hired as a security guard just four days earlier, an official said Friday.
The would-be assassin, who was shot dead by Karzai's American bodyguards, was from neighboring Helmand province, a former stronghold of the ousted Taliban.
A bystander and one of Karzai's Afghan bodyguards were also killed in the melee, according to Khalid Pashtoon, spokesman for Kandahar Gov. Gul Agha Sherzai.
Television pictures showed a bullet hole in the back of Karzai's seat and another in the car window alongside. The BBC said the second bullet narrowly missed Karzai's face.
The explosion Thursday was the worst act of violence the wrecked city has seen since the hard-line Taliban regime was ousted in fighting last year.
Police Chief Basir Salangi said 26 people were killed and 150 wounded in Thursday's blast in a crowded bazaar near the Information Ministry in downtown Kabul, amending his earlier assessment putting the death toll at 10.
Salangi said two men were detained for questioning in the bombing. Both previously owned the taxi in which the bomb was placed, but the latest owner of the car has not been found.
International peacekeepers wearing flak jackets moved through the city in small convoys of armored personnel carriers and jeeps, stopping at intersections to inspect passing cars.
Afghan police and soldiers armed with assault rifles erected roadblocks all across Kabul — in some areas every few hundred yards.
Soldiers stopped cars and taxis, peering into back seats and trunks looking for explosive material or anything deemed suspicious. Many streets were blocked off altogether with troops waving back lines of traffic.
Salangi said Karzai was due to meet Russian Defense Minister Sergei Ivanov, who is in Kabul for a one-day visit. Salangi said security had been stepped up in preparation for the afternoon meeting.
"We've got 800 extra soldiers deployed throughout the city. They're at every intersection, checking cars for bombs," Salangi said.
Thursday's bombing in the capital and the failed assassination attempt against Karzai in the southern city of Kandahar amounted to the most serious threat to the transitional government since it took office in June. The fledgling administration is struggling to bring order and security to a country wracked by decades of bloodshed.
Afghan officials blamed Osama bin Laden's al Qaeda network for the violence, but offered no proof to back the allegations.
"Everybody is angry with al Qaeda," said 29-year-old Mul Tan, who was driving a truck at the site which workers were filling with debris. "This is not a military site. Look at all these shops. They wanted to kill all these people."
High on a hill overlooking Kabul, a father helped bury his dead son — a moneychanger who succumbed to wounds overnight at Wazir Akbar Khan hospital.
"I had two sons before, now I only have one," Shah Mohammad said, weeping quietly as a dozen men removed their turbans, stretching them under the still-bloody body to lower it into a fresh, rocky grave.
Authorities said they had been bracing for possible terrorist violence ahead of the first anniversary of the Sept. 11 attacks in the United States, and of the Sept. 9 assassination of Ahmed Shah Massood, the revered military commander of the northern alliance. Massood's slaying resonates even more in Afghanistan than the watershed attacks in New York and Washington two days later.
Massood, considered a hero of the resistance of the 10-year Soviet occupation in the 1980s, led the battle against the Taliban regime for five years that ended last November with the help of a massive U.S. bombing campaign. His murder by suicide bombers was blamed on bin Laden's network.
Karzai is scheduled to make a series of public appearances to mark the anniversary of the famous Afghan commander's death, reports Logan, which will make tighter security more difficult.