Elon Musk said one of Twitter Inc.’s top lawyers “was exited,” part of the fallout from the billionaire’s unusual efforts to release internal communications to criticize prior practices at the company he bought almost six weeks ago.

Mr. Musk announced in a tweet Tuesday the departure of Deputy General Counsel Jim Baker, who had been with the company since 2020. Since completing the $44 billion acquisition, Mr. Musk has fired several company leaders, including Mr. Baker’s immediate boss. 

Mr. Baker didn’t respond to a request for comment Tuesday. 

Mr. Musk’s tweets Tuesday linked the departure to a project he has dubbed the “Twitter Files”—the release of internal communications that he has characterized as showing biased decisions by previous management benefiting Democrats. Mr. Musk said he gave journalists Matt Taibbi and Bari Weiss access to company documents in an effort to rebuild public trust in Twitter. Both publish on Substack, which lets writers publish and monetize their own newsletters.

“Why should people believe Twitter in the future if Twitter does not come clean about the past?” Mr. Musk has said.

Releasing in-house messages to outside scrutiny is another example of the unorthodox measures taken by Mr. Musk in his management of Twitter. Employees have faced weeks of turmoil, including the firing of half of Twitter’s staff and abrupt decisions about product plans.

Silicon Valley has seen numerous leaks of internal company information, but generally from employees unhappy with management. Mr. Musk has turned that on its head.

The initial batch of documents made public Friday revealed the names of rank-and-file employees involved in content moderation, which Twitter’s former head of trust and safety, Yoel Roth, warned “puts them in harm’s way and is a fundamentally unacceptable thing to do.”  

Mr. Musk afterward said email addresses should have been excluded.

In recent days, Mr. Musk has said he was motivated to buy Twitter over concerns that its content moderation silenced voices. Mr. Musk has said he thinks of Twitter as the digital town square. Since taking over the platform, he has pulled several levers to make sure the conversation on Twitter is also about Twitter. He reinstated the account of former President Donald Trump, urged users to flock to Twitter to follow the soccer World Cup, and, at least for a brief period, picked a fight with Apple Inc.

“So many interesting posts on Twitter these days!“ he tweeted Monday to his roughly 120 million followers.

Mr. Musk publicized the exit of the deputy general counsel on Tuesday, expressing “concerns about Baker’s possible role in suppression of information important to the public dialogue,” without providing more details. Mr. Taibbi said Mr. Baker was vetting documents given to himself and Ms. Weiss. Twitter didn’t respond to a request for comment.

Before Twitter, Mr. Baker held various roles at the Justice Department, including as general counsel for the Federal Bureau of Investigation from 2014 to 2018.

There is some precedent for companies that are dealing with controversy to performs high-profile self-examinations, though they usually hire outside law firms or former government officials rather than tapping journalists to review the materials.

In 2014, General Motors Co. Chief Executive Mary Barra hired a former U.S. attorney to chronicle why it took the auto maker 11 years to recall cars equipped with defective ignition switches. The more-than-300-page report was made public and named names, including leveling blistering criticism against a midlevel engineer. 

Mr. Musk approached the matter in his own way, teasing the Twitter Files to his followers ahead of time. 

On Friday, he pointed readers to a string of tweets by journalist Mr. Taibbi, who spent about two hours posting about records that dealt with Twitter’s controversial decision shortly before the 2020 U.S. presidential election to curb sharing of New York Post articles about Hunter Biden, son of now-President Joe Biden.

The tweets included screenshots of what Mr. Taibbi said were internal communications in which executives including Mr. Baker deliberated over that decision. Facebook also initially limited the distribution of the New York Post articles.

The Post and The Wall Street Journal are both owned by News Corp.

In explaining its handling of the Post story, Twitter initially cited a potential violation of its rules regarding hacked materials. Within two days, it reversed itself and was allowing the story to be linked in tweets. Twitter later unlocked the Post’s account after a two-week standoff, and executives have since said the initial decisions were wrong. Asked about the Hunter Biden story while testifying before Congress last year, Jack Dorsey, CEO at the time of the decision, said Twitter didn’t base its policies on any political leanings, though he also said the company made a “total mistake” regarding the Post articles. 

Twitter’s and Facebook’s restrictions reduced the spread of the articles, but didn’t stop links to them from reaching an audience on their platforms outright. Some data scientists have suggested the bans helped draw more attention to the Post article. 

The Biden campaign said at the time that the New York Post never asked the campaign about the critical elements of the story before publishing and that investigations found no wrongdoing by Joe Biden.

The internal exchanges released Friday appeared to some people involved in content moderation as typical discussions between employees racing to figure how to handle a difficult policy issue.

Some conservatives, such as Rep. Kevin McCarthy (R., Calif.), embraced Friday’s document release as evidence for claims that major social-media platforms are biased against the political right. Twitter’s own researchers said in a report last year that the platform’s algorithms amplified voices on the political right in several countries, including the U.S.

Mr. Taibbi said in a tweet on Tuesday that more information from the data Twitter has handed over would be released. He has said he “had to agree to certain conditions” in exchange for the opportunity to write about the Twitter documents, but didn’t elaborate. Mr. Taibbi couldn’t immediately be reached for comment Tuesday. 

Mr. Musk has suggested he might provide “a data dump” to the public at some point after the “critical stories” have been told. 

Write to Alexa Corse at alexa.corse@wsj.com and Tim Higgins at tim.higgins@wsj.com