LONDON — Seeking to quell a growing uproar over its mishandling and possible cover-up of sexual abuse allegations, the British Broadcasting Corporation broadcast a highly critical examination Monday night of what one insider called the worst crisis in half a century to afflict the BBC, a national institution once viewed as an emblem of ethical probity.

The unusual spectacle of one BBC show broadcasting a hard-edged look at the journalistic integrity of another reflected the sense of crisis across Britain, as journalists, school officials, politicians and others question whether they did enough to stop decades of widely-rumored abuse by Jimmy Savile, a high-profile television personality and disc jockey known also for his philanthropy, who is accused of victimizing some 200 girls.

Previous BBC associates of Mr. Savile who appeared on the “Panorama” investigative program spoke of their suspicions about him — even of his seeming boasting about what he had done — and of their failure to expose him.

Bob Langley, a former reporter who covered a charity run in which Mr Savile participated, said he saw girls of “12 or 13, they could have been 14” emerging from Mr. Savile’s trailer at the event. “After they had gone he indicated to me in a nudge, nudge, wink, wink sort of way that he had just had sex with them,” Mr. Langley said.