Updated

The United States encountered mounting challenges to its presence in Afghanistan Thursday, as Afghan President Hamid Karzai urged the U.S. to pull back to military bases and the Taliban announced they were suspending tentative peace talks.

The developments come after two incidents strained relations between the two countries -- the inadvertent burning of Korans on a U.S. base, and most recently a shooting spree that left 16 Afghan civilians dead. Afghan lawmakers expressed outrage after the U.S. soldier suspected in that massacre was flown out of the country to Kuwait.

By Thursday, both Karzai and the Taliban were lashing out at American negotiators, all while Defense Secretary Leon Panetta was visiting the country.

Karzai, in a meeting with Panetta, asked the U.S. to withdraw from Afghan villages and stay on bases, saying, "Afghan security forces have the ability to keep the security in rural areas and in villages on their own."

He also urged the U.S. to let Afghan forces take the lead for countrywide security in 2013, a year ahead of schedule.

More On This...

Shortly afterward, the Taliban announced they were breaking away from talks with the U.S. because the U.S. "turned back on its promises." One concession the Taliban were looking to extract out of peace talks was the release of prisoners from Guantanamo Bay.

The statement said the Taliban would suspend talks "until the Americans clarify their stance on the issues concerned and until they show willingness in carrying out their promises instead of wasting time."

White House Press Secretary Jay Carney, at Thursday's briefing, would not comment in detail on the Taliban talks other than to say the U.S. supports an "Afghan-led process towards reconciliation."

As for Karzai's comments, Carney said the plan remains to fully transfer security over to the Afghans in 2014.

"That's the mission and we are sticking to it," Carney said.

The Karzai statement could be problematic. Karzai told Panetta that the weekend shootings in southern Afghanistan were cruel and that everything must be done to prevent any such incidents in the future. He said that was the reason he was demanding the pullout from rural areas and early transfer of security.

Currently, U.S. and NATO forces are scheduled to mostly withdraw combat forces by the end of 2014.

Panetta meanwhile shrugged off a potentially lethal attack on a coalition base in Afghanistan as his plane was about to land, insisting he was not the target.

"I have absolutely no reason to believe that this was directed at me," Panetta said after meeting Karzai.

Panetta said he was flying into a war area, and therefore "we're going to get these kind of incidents."

An Afghan man carrying gasoline and a lighter drove a stolen truck onto the tarmac at Camp Bastion in Helmand as Panetta's plane came in to land Wednesday. After crashing into a ditch, he lit himself on fire.

The attacker, identified by the Wall Street Journal as a contract interpreter, was taken to a military hospital with burns over most of his body and died Thursday.

A British serviceman was injured in the incident, with the attacker then driving toward a group of U.S. Marines, who gathered to greet Panetta on the runway ramp.

"My personal opinion is yes, he had an attempt to harm," U.S. Army Lt. Gen. Curtis M. Scaparrotti, the day-to-day military commander in Afghanistan, said. "I think, frankly, he tried to hit the people on the ramp."

About 1,000 Afghans took to the streets Thursday to protest the mass killing of 16 civilians and the earlier unintentional burning of Korans by U.S. soldiers, police said.

The protesters were demanding a public trial for the shooter, identified as a 38-year-old U.S. Army sergeant who has since been flown out of Afghanistan.

Army sources told Fox News said he was in Kuwait. Pentagon spokesman Capt. John Kirby said the decision to remove him was made because there was no appropriate detention facility inside the country.

For more information on U.S. troops in Afghanistan, see The Wall Street Journal

Newscore and The Associated Press contributed to this report.