Deepening the political turmoil that has engulfed Britain’s public broadcaster, the chairman of the BBC, Richard Sharp, resigned on Friday after an investigation concluded that he failed to disclose his involvement in arranging a nearly $1 million loan for Boris Johnson while he was prime minister. Mr. Sharp, a former banker at Goldman Sachs and major donor to Mr. Johnson’s Conservative Party, said in a video statement before the report’s release that the omission was “inadvertent and not material,” but that he had decided to step down from the broadcaster’s board to “prioritize the interests of the BBC.” His departure intensifies the pressure on the BBC, a revered institution that once anchored Britain’s media with its reliable public funding and commitment to impartial journalism, but now seems to lurch from crisis to crisis. In an era of polarized politics and freewheeling social media, the broadcaster has found itself in the cross hairs from both the left and right — whether for its news coverage, which Conservatives say violates its credo of political impartiality, or for its recent internal decision-making, which some on the left say is influenced by not wanting to offend the Conservative lawmakers in power. Last month, the BBC ignited a national firestorm by suspending Gary Lineker, a former soccer star and its most prominent broadcaster, after he went on Twitter to liken the government’s immigration policy to that of Germany in the 1930s. His suspension triggered a walkout of much of the BBC’s sports staff in solidarity with Mr. Lineker, and forced it to broadcast “Match of the Day,” its flagship weekly soccer program, without commentary. The BBC settled the dispute by vowing to review its policies governing the use of social media by its on-air personalities. But the episode threw a harsh spotlight on Mr. Sharp because he refused to step down from his post, even though he was being investigated over his role in the loan to Mr. Johnson. His compromised position also meant he could not defend the BBC at a time when it was coming under fierce criticism — both for the haphazard enforcement of its social media guidelines, as well as for the settlement with Mr. Lineker, which many Conservatives derided as a capitulation. That Mr. Sharp, 67, played a role in shoring up the personal finances of a prime minister has added to the perception of a conflict of interest in the governance of Britain’s most important media institution. In the 25-page report, Adam Heppinstall, a barrister, identified two cases that he said “gave rise” to potential perceived conflicts. In one, he said Mr. Sharp told Mr. Johnson of his plan to apply for the post of BBC chairman before he submitted a formal application in November 2020. In the other, more serious case, Mr. Sharp told Mr. Johnson that he planned to introduce a wealthy Canadian businessman, Sam Blyth, to a senior government official, who could help arrange a loan for Mr. Johnson. Mr. Blyth, a distant cousin of Mr. Johnson, had volunteered to act as guarantor for the loan of 800,000 pounds ($996,000) to the prime minister, whose finances were being strained by a recent divorce and a new family. Mr. Sharp’s involvement did not extend further than that. But Mr. Heppinstall concluded there was a risk that his selection for the chairman’s post would be perceived as a reward for his assistance to Mr. Johnson. Mr. Heppinstall also cited the risk that Mr. Sharp would not be viewed as independent once in the job. His investigation did not assess the propriety of Mr. Johnson seeking the loan while in office. Mr. Johnson’s finances were frequently under scrutiny during his turbulent three years in Downing Street. He paid for a lavish refurbishment of his official apartment with funds from another party donor, prompting an outcry that motivated him to pick up the tab himself. In an interview with Sky News in January, Mr. Johnson said Mr. Sharp “knows absolutely nothing about my personal finances.” Mr. Johnson, who has long derided the BBC for its political correctness, dismissed the matter as “just another example of the BBC disappearing up its own fundament.” Still, the reports of Mr. Sharp’s involvement in the loan prompted calls by the opposition Labour Party for his resignation. The current prime minister, Rishi Sunak, distanced himself from Mr. Sharp, even though the two had once worked together at Goldman, where Mr. Sunak was a young banker before going into politics. On Friday, a spokesman for Downing Street said Mr. Sharp’s decision to resign was “a matter for him and the BBC.” In his statement, Mr. Sharp said he was gratified that the report confirmed the limited nature of his involvement in the loan. But he said he regretted not raising it with an appointments panel before he took the post of chairman. “I wish, with the benefit of hindsight, this potential perceived conflict of interest was something I had considered to mention,” Mr. Sharp said. “I would like once again to apologize for that oversight — inadvertent though it was — and for the distraction these events have caused the BBC.” For the broadcaster, the resignation is the latest in a string of episodes that, depending on one’s political vantage point, raise questions about its impartiality or its determination not to antagonize an already hostile government. In 2020, the BBC announced it would strip lyrics from two well-known patriotic songs during an annual televised concert, drawing outrage from Mr. Johnson. The lyrics, some said, evoked a British colonial past and were at odds with the Black Lives Matter movement then sweeping the Western nations. The BBC later reversed the decision. In 2021, the government assailed the BBC after one of its hosts mocked a cabinet minister for appearing in an interview with a large Union Jack behind him. A few days later, the government decreed that the flag should fly on all public buildings every day of the year, rather than simply on designated days. The New York Times