The BBC has been embroiled in a spate of allegations that it had failed to maintain political neutrality in its reporting including in its coverage of Trump, the Israel-Hamas war and over transgender issues.
"This is entirely my decision, and I remain very thankful to the chair and board for their unswerving and unanimous support throughout my entire tenure, including during recent days," Davie said in a statement.
"I have been reflecting on the very intense personal and professional demands of managing this role over many years in these febrile times, combined with the fact that I want to give a successor time to help shape the charter plans they will be delivering.
"Overall the BBC is delivering well but there have been some mistakes made and as director-general I have to take ultimate responsibility," he said.
The Daily Telegraph had reported for days on an internal document produced by a former BBC adviser on standards who had listed a raft of errors, including in the way a speech by Trump on January 6, 2021, was edited.
The document indicated the Panorama program had edited two parts of Trump's speech together so that he appeared to directly encourage the Capitol Hill riots of January 2021 - cutting out a section where Trump said that he wanted supporters to demonstrate peacefully.
The Panorama program, titled Trump: A Second Chance?, was broadcast by the BBC the week before last year's US election.
Turness said the controversy about the Trump documentary "has reached a stage where it is causing damage to the BBC - an institution that I love. As the CEO of BBC News and Current Affairs, the buck stops with me."
Trump in the BBC program was shown telling his supporters that "we're going to walk down to the Capitol" and that they would "fight like hell," a comment he made in a different part of his speech.
White House press secretary Karoline Leavitt reacted on X, posting a screen grab of an article headlined "Trump goes to war with 'fake news' BBC" beside another about Davie's resignation, with the words "shot" and "chaser."
Davie will stay on for the next few months while a replacement is found.
A person familiar with the situation said Davie's decision had left the BBC board stunned by the move.
Pressure on the broadcaster's top executives has been growing since the Daily Telegraph newspaper published parts of a dossier complied by Michael Prescott, who had been hired to advise the BBC on standards and guidelines.
As well as the Trump edit, it criticised the BBC's coverage of transgender issues and raised concerns of anti-Israel bias in the BBC's Arabic service.
Widely respected around the world, the BBC has in recent years been accused of failing to maintain its commitment to impartial news, struggling to navigate the deeply polarised political and social environment.
The BBC faces greater scrutiny than other broadcasters because of its status as a UK institution funded through an annual licence fee of STG174.50 ($A352) paid by all households with a television.
It also is bound by the terms of its charter to be impartial in its output.
with AP and PA
