David Cameron has accepted that Afghanistan will not be "a perfect democracy" by the time British troops return home, but said that the public wants to see an "endgame" to a military operation that has lasted more than a decade.
The prime minister was speaking ahead of talks with President Barack Obama in which the two leaders are expected to sketch out a timetable for the withdrawal of international troops from Afghanistan by the end of 2014.
After a red carpet welcome at Andrews Air Base on Tuesday, the Prime Minister joined Mr Obama on board Air Force One for a 70-minute flight to Ohio to enjoy a university basketball match ahead of Wednesday's talks in the White House.
No announcement of further troop drawdowns is expected during his three-day visit, but the two leaders will seek to identify a date - possibly as early as mid-2013 - for Afghan forces to take over the lead security role throughout the country, the Press Association reported.
The transition date is expected to be formally agreed and announced at Nato's summit in Chicago in May, and will allow British and American troops to move into a support role and pave the way for the return home of large numbers in a gradual process beginning next year.
Mr Cameron and Mr Obama used a joint article in the Washington Post to say they were "proud of the progress our troops have made in dismantling al Qaida, breaking the Taliban's momentum and training Afghan forces".
In the wake of the recent deaths of six British soldiers and the massacre of 16 Afghan civilians by a renegade US serviceman, they acknowledged that Nato-led operations in Afghanistan remain "a difficult mission".
But both men indicated they would not be knocked off course or pushed into a precipitate exit from Afghanistan, saying: "We honour the profound sacrifices of our forces and in their name we'll carry on the mission."
Mr Cameron made clear that his view of what constitutes a successful completion to Britain's mission is less ambitious than the dream of a modern democracy in the heart of central Asia bandied around in some quarters when the Taliban regime was first toppled in 2001.
"What I define as doing the job is leaving Afghanistan looking after its own security, not being a haven for terror, without the involvement of foreign troops," the Prime Minister told reporters.
"That should be our goal. So that the British public, our troops and the Afghan government frankly, know there's an end to this.
"This is what we defined in 2010 through the National Security Council. I accept it won't be a perfect democracy. There will be huge development problems."
In recognition of growing public weariness with the war, which a large majority of voters in a recent poll said was unwinnable, Mr Cameron added: "I think people want an endgame. They want to know that our troops are going to come home, they have been there a very long time."
Mr Cameron said that he and Mr Obama will use this week's talks to ensure that the UK and US are "absolutely in lock-step" over the withdrawal process, as well as to discuss future support for Afghanistan after the International Security and Assistance Force (Isaf) has left.
At the request of President Hamid Karzai, Britain has agreed to set up and staff a Sandhurst-style military academy for Afghan army officers, and will also provide funding for training security forces.
Syria is also high on their agenda, with both Mr Cameron and the President admitting frustration at the ongoing bloodshed in the city of Homs as dictator Bashar Assad cracks down on centres of opposition to his regime.
They will look at what additional diplomatic and economic pressure can be brought to bear on Damascus to speed up Assad's removal from power, but are not thought likely to give consideration to more radical measures such as arming rebel groups or providing air cover of the kind seen in Libya last year.
"We want transition," said Mr Cameron. "We want Assad to go. That is the shortest route to stop the bloodshed.
"We are both very frustrated by the situation and wanting to do more in working out how we can add to the pressure the regime faces."
In their joint article, Cameron and Obama issued a warning to Iran's president Mahmoud Ahmadinejad to fulfil international obligations on his nuclear programme or "face the consequences".
But while both men have made clear they are not ruling out any course of action, they insisted that there was still plenty of scope for increasing pressure on Tehran, including through US sanctions due to come into effect later this year which will effectively blacklist anyone who has dealings with Iran's national bank.
"We both believe there is a longer road of sanctions and pressure to run, but nothing should be taken off the table," said Mr Cameron.
Tuesday's visit to the basketball game between Mississippi Valley State University and Western Kentucky University is part of a packed programme designed to display the warmth of trans-Atlantic relations, including a star-studded state dinner in the White House on Wednesday night.
Mr Cameron is the first foreign leader to be invited by President Obama on board his personal plane, and last evening said he had no doubt of the continuing strength of the special relationship.
"I think the special relationship survives. It's increasingly strong, based on common interests and common values," said the PM.
"We shouldn't have to take its temperature all the time. Sometimes we can over-analyse how many phone calls and how many meetings, but I don't look at it like that."
Mr Cameron said he was "very glad I do get on so well with Barack Obama", adding: "Obama's approach is deeply rational and reasonable, and also very strong."
Basketball is one of the president's passions, and he famously had a court installed at the White House so he could continue to play regularly while in office.
The game marked the start of the hugely popular College March Madness tournament, which is televised to large audiences across the US.