"These six days have been the most productive of the rounds we've had with the Talibs", Khalilzad said in Qatar.
"For the first time, I can say we have had substantive discussions, negotiations, and progress on all four issues".
That engagement has been put on hold for the two-day Afghan dialogue and will resume on Tuesday, both sides have said.
"The Taliban should know that they can not gain leverage by the killing of civilians", Ghani said in a statement. "We have not faced any obstacles yet".
The Afghan delegation includes various stakeholders, including former mujahedeen (Islamic warriors) who fought the Soviet Union in the 1980s, as well as former government officials, former ambassadors, civil society representatives and a small number of women.
But the Taliban have refused to negotiate with the US -backed Afghan government, denouncing it as a U.S puppet. "A great success would be for them to come up with a framework that could lead into direct negotiations, Afghan-Afghan, and hopefully catch up with the speed in which the talks between the Taliban and the United States are progressing", said Sultan Barakat, the director of the Center for Conflict and Humanitarian Studies at Doha Institute, who has been closely involved in organizing the event.
The Taleban, who have repeatedly refused to negotiate with the Western-backed government of Afghan President Ashraf Ghani, agreed to join the intra-Afghan summit on the condition that those there would attend in a personal capacity.
Khalilzad said it will ultimately be up to Afghans to decide among themselves the agenda for negotiations as well as the terms of a ceasefire.More news: Kiwi cyclist George Bennett wins team stage at Tour de France
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Ghani s administration, which the Taliban consider a puppet regime, has also been excluded from the direct US-Taliban talks.
Abdul Hai Khateby, a spokesman for the provincial governor, said the Taliban planted the mine apparently to thwart a planned Afghan military offensive to retake nearby areas under the militants' control.
He said Washington's "aspiration" is to have that framework in place by September 1 and ahead of the Afghan presidential election.
Jamila Afghani, a prominent women's rights activist, was getting ready to address a peace dialogue attended by Afghan leaders and Taliban representatives in Doha, when she got news of an attack back home.
Trump has overturned USA policy on Afghanistan's neighbour Iran, previous year pulling out of a nuclear deal Tehran signed with world powers.
Khalilzad said the policy of Pakistan, which Kabul and Washington have long accused of backing the Taliban, has changed "little-by-little in a positive direction".
"The Iran deal is not the first deal we've got out of", said Khalilzad.
"I think there is a strong possibility that there could be an agreement between the U.S. and the Taliban even sooner than September, but an agreement that is just between the USA and the Taliban is not a peace agreement for Afghanistan", said Laurel Miller, who served as the United States special representative for Afghanistan and Pakistan under both Trump and his predecessor Barack Obama.