The BBC’s director general has expressed his exasperation with “conspiracy theories” about the broadcaster’s election news coverage, although some of its journalists privately fear that errors during the campaign may have hit public trust in the corporation.

Tony Hall emailed staff on Friday to thank them for their work on the BBC’s coverage, which has led to the corporation’s political news output coming under intense online scrutiny. This followed criticism of the editing out laughter aimed at Boris Johnson in a news bulletin, reporters uncritically repeating Conservative sources, such as when a Labour activist was erroneously accused of punching a Tory aide, and the prime minister escaping scrutiny after dodging a one-on-one interview with Andrew Neil.

Hall said the BBC’s critics were often seeing bias in what were genuine human errors: “In a frenetic campaign where we’ve produced hundreds of hours of output, of course we’ve made the odd mistake and we’ve held up our hands to them. Editors are making tough calls every minute of the day. But I don’t accept the view of those critics who jump on a handful of examples to suggest we’re somehow biased one way or the other.”

He also suggested social media platforms should find ways to reduce the level of public criticism aimed at journalists, such as BBC political editor Laura Kuenssberg: “Elections always put the BBC’s impartiality in the spotlight. Social media offers a megaphone to those who want to attack us and makes this pressure greater than ever. The conspiracy theories that abound are frustrating. And let’s be clear – some of the abuse which is directed at our journalists who are doing their best for audiences day in, day out is sickening. It shouldn’t happen. And I think it’s something social media platforms really need to do more about.”

Despite this, there remain concerns within the corporation’s newsrooms that this election was a tough challenge for a broadcaster that strives for accuracy. More than 30 BBC journalists spoke to the Guardian about the outlet’s coverage in the final days of the campaign, ranging from senior on-air presenters to mid-level producers and recent hires.

Almost all of them, who spoke anonymously to give their frank views, believed strongly in the BBC’s purpose and importance as a unifying force of reliable news in a divided country. But many raised doubts as to how the corporation’s output can withstand such a level of scrutiny. A common complaint was the risk of feeling paralysed by the knowledge that every single piece of television, radio or online output is at risk of being examined in detail on Twitter for allegations of bias.

While older staff were more likely to dismiss online complaints about coverage, younger staff repeatedly and persistently raised concerns about the loosely phrased tweets of senior BBC journalists, which often reach millions of people without passing through editors.

“They should be subject to the same rigour as any other online material,” one journalist said.

Another employee involved in the broadcaster’s politics output during the election said: “I’m proud of the programmes we’ve put together, but I feel like we’ve been undermined at every turn by constant gaffes on the part of senior presenters and editors. I detect an unhealthy us-versus-them mentality, an unwillingness to say sorry when the BBC gets it wrong and a genuine terror of upsetting the government in particular.”

There was repeated concern that the BBC’s model of neutral reporting was not suited to an age where politicians are more willing to try to manipulate the media, with Johnson and his aide Dominic Cummings repeatedly singled out for taking advantage of that.

Fran Unsworth, director of news and current affairs, told the Guardian that the broadcaster faced a challenge to fight back against claims of bias: “Things go viral very quickly and our ability to respond and deal with them is limited. Even when we try to explain and put out the denials or the explanation as to how things might’ve happened. People just aren’t buying it. It’s a real issue for us to deal with it.”

While the BBC has long been a punchbag of right-wing publications such as The Sun and Daily Mail, there has been much more criticism from left-wing campaigners and centrist pro-Remain campaigners this election. BBC journalists based in London, an area with strong Labour support, reported finding it harder now that their friendship groups were increasingly critical of the corporation’s output.

Many complained that they had come to dread discussing their jobs at family events or children’s parties because they would find themselves cornered and cross-examined on accusations of BBC bias.

One regional BBC reporter said they had noticed many more members of the public refusing to talk to him, although it was unclear whether this was just part of a general anti-media reaction among the public. “It felt like BBC reputation has taken a bit of a hit and not just in the Twitter vacuum. Not that I think there is a bias of course. Just a series of errors have unfortunately fallen a similar way, which gives plenty of ammo.”

Although many of the more prominent anti-BBC complaints in this election have come from the left of politics, staff pointed out that pro-Conservative Brexit supporters often file more complaints about coverage directly to the corporation.

One BBC journalist working on a high-profile radio programme expressed exasperation with “ludicrous Twitter storms” over minor errors but admitted criticism had made life harder. “It’s really tough working for the BBC at the moment. I have never known the fury to be so great. But my overwhelming impression is that people should be careful what they wish for, because they will end up with Fox News or Russia Today.”

Unsworth said it was not realistic to ask correspondents to step back from Twitter but there is room for changes: “It is an important form of communication to get their stories out. We just need to reinforce our social media rules. But I don’t think it’s viable to say take a step back.”

She also said she sees her duty as being to the whole of the BBC’s audience, not just political activists scrutinising the BBC’s output on social media. “This is a smaller world than the people think it is. I find people who are living ordinary lives and they’re not on social media every day. They are still taking their cues from the mainstream news.”

Unsworth said she had been proud of the corporation’s output during the election and preferred scrutiny rather than being ignored.

“It shows that we matter. It would be worse if they weren’t talking about us. All the talk had been that the mainstream media doesn’t really matter anymore.”