David Cameron has admitted that the Syrian president, Bashar al-Assad, has strengthened his position in recent months, and warned that the country faces a "depressing trajectory".

In an interview on the Andrew Marr Show on BBC1, the prime minister gave his clearest indication to date that Britain will not be supplying arms to the Syrian rebels despite pressing for the lifting of the EU arms embargo.

Cameron insisted he was still committed to helping the Syrian opposition, but admitted its numbers included "a lot of bad guys". He also acknowledged that Assad had strengthened his position.

The prime minister said: "I think he may be stronger than he was a few months ago, but I'd still describe the situation as a stalemate. And yes, you do have problems with part of the opposition that is extreme, that we should have nothing to do with."

But Cameron said it would be wrong to abandon the opposition, although he indicated Britain would provide only non-lethal equipment.

He said: "[Having extremists in the opposition] is not a reason for just pulling up the drawbridge, putting our head in the sand – to mix my metaphors – and doing nothing. What we should be doing is working with international partners to help the millions of Syrians who want to have a free democratic Syria, who want to see that country has some chance of success."

Asked about arming the opposition, the prime minister said: "We're not intervening by supplying weapons, but I think we can with partners … to strengthen those parts of the Syrian opposition that really do represent the Syrian people."

Cameron denied his wife, Samantha – who was deeply moved by the plight of Syrian refugees when she visited a refugee camp in Lebanon – was dictating government policy. The Times reported last week that Samantha Cameron had been pressing for greater intervention.

"It's a total urban myth," Cameron said of the claims. "Samantha went to a refugee camp in Lebanon because she wanted to see for herself. She does some work for Save the Children and was very moved by what she saw. You hear from people – and I've done this – who are fleeing an appalling regime that is murdering their husbands, wives, children, wrecking their communities.

"The point that she's made, and that I've made, is also that the regime is fomenting the extremists, so they can say to the rest of the world, look, the alternative to me, Assad, is these appalling extremists."

The prime minister added: "She does not influence my policy on this. I've been very passionate about this for a long time. But I would accept that we are on a depressing trajectory and we need to change that."

The UK-based Syrian Observatory for Human Rights and the Local Coordination Committees, two opposition groups tracking the violence in Syria, said at least 20 people were killed in the northern town of Ariha, including two children and two women. It was not immediately clear what triggered the shelling. State media said on Sunday that government forces killed nearly 50 rebels in an ambush near Damascus. The Guardian cannot independently verify the figures.