Media bias is always in the eyes of the beholder. So, on one side, those who support leaving the EU regard the BBC as the Brussels Broadcasting Corporation. Those who seek to remain, meanwhile, view it as the Brexit Broadcasting Corporation.

This is familiar ground for Britain’s major public service broadcaster. Think back to the Troubles, the Falklands war, the Scottish independence referendum. In each case, the BBC was accused of failing to be impartial.

This time, however, the attacks pose a more profound challenge. At a time when fewer and fewer people believe what they are told by journalists, the BBC is struggling to convince its audience that it is not subject to sinister influences. The public’s media savviness has gradually elided into media cynicism. People are on red alert for bias. They have come to suspect misinformation and propaganda where it does not necessarily exist.

Brexit has seen a marked rise in complaints to the BBC, mostly from the pro-leave side. Audience surveys carried out on behalf of the corporation reveal a worryingly negative reaction to its coverage. And a review of BBC news and current affairs by the regulator, Ofcom, released last week, mentioned “dissatisfaction with the BBC’s coverage of Brexit”, although it did not record specific examples.

The criticism is rooted, partly, in the intense polarisation of each side. Positions have become ever more entrenched, with leavers fearing that a pro-remain establishment, which is thought to include the corporation, is conspiring to ensure that the referendum result is not honoured.

Another major factor behind the escalating hostility towards the BBC is the saga’s longevity. Brexit just won’t get done and, as it drags on, the corporation’s detailed daily coverage opens its content to increased scrutiny, and increased scepticism, from viewers and listeners.

As Anne McElvoy has pointed out, in a column urging the corporation to underline its commitment to impartiality, “the strains of Brexit” have made it tricky for BBC journalists to reflect the balance between two fiercely held viewpoints. Audiences do not seem to believe that news editors and their staff take pains to hold the middle ground.

Doubtless, people are also influenced by the critics who rage against the BBC’s output. Aside from the background hubbub of anti-corporation bile from the usual suspects – newspapers such as the Sun, Daily Mail and Daily Telegraph – there have been specific assaults. Steve Baker, the Tory MP who chairs the European Research Group, accused BBC journalists of asking questions based on false premises.

Paul Dacre, the former Mail editor, was as forthright as ever in writing of “the Eurosceptic-hating BBC” which “pumps out hysterical anti-Brexit propaganda”. In a column for the Spectator he called its coverage “a disgrace” and singled out its Europe editor, Katya Adler, for special mention, asking if she is “actually employed by Brussels”.

This is a silly point. When Adler goes on air, she says what her EU sources have told her. Those sources are anything but happy about Brexit and the UK’s handling of the crisis, so it is hardly surprising that her reports reflect that. How, in such circumstances, could she frame her comments differently?

Charles Moore, the Telegraph columnist, opened another line of attack back in April when he complained on Question Time about being the sole leave supporter on its panel. This was a specious objection – another guest, the culture secretary, Jeremy Wright, had backed remain in 2016 but now favours leaving the EU. Moore’s criticism rested, in part, on a flawed study by the Institute for Economic Affairs, which counts panellists on BBC discussion shows based on how they voted in the referendum, rather than their current views on Brexit.

Remainers have their own complaints. In this column last week, Emily Bell contended that the BBC’s Brexit coverage lacked “adequate authority”. She added that its journalism has “struggled to be serious and consistent enough to meet the complex gravity of the moment”.

I beg to differ. There have certainly been moments when I’ve shouted at the TV and radio because interviewers have failed to ask the question that I felt was essential. But I often have the same reaction to non-Brexit matters. Those cavils aside, the coverage has been informative, if repetitive.

Perhaps the most contentious criticism of late has come from Peter Oborne, the journalist who campaigned and voted for Brexit and then, six months ago, changed his mind. In a scathing analysis of Brexit reporting, in which he takes several political editors to task for acting as stenographers for fabricated stories, he included a section on the BBC’s political editor, Laura Kuenssberg.

His point, somewhat similar to Dacre’s criticism of Adler, centres on her use of anonymous sources. Oborne seized on Kuenssberg’s posting of six linked tweets as proof that she was being “manipulated by Downing Street”.

Why? Because she conveyed that Boris Johnson’s government was holding “two positions” on the Benn act. On the one hand, it would comply with it. On the other, it was determined not to do so.

Kuenssberg’s insight relied, as she stated in her tweets, on “a senior No 10 source”. That didn’t strike me as unusual, nor did I think it misleading. It was not a lie.

Johnson was, at the time, trying to have his cake and eat it. Kuenssberg did nothing more than reflect his ambivalence. As for her use of an unnamed inside source, it is simply how lobby journalism works. But it doesn’t have to be that way. I agree with Oborne: it is time for journalists to be more forthcoming in naming the people we quote.

None of that should detract from Kuenssberg’s fine work, day after day. Fair-minded BBC viewers and listeners may well agree with me. But can we be sure that they are in the majority?