Washington (CNN) Top White House adviser Kellyanne Conway has apologized for citing a fake "massacre" in support of Donald Trump's travel ban, in an interview on CNN in which she acknowledged that not all of the mainstream media can fairly be characterized as "fake news".

Conway said "regretted tremendously" shaming the press for not covering the "Bowling Green massacre", an event that did not happen, when she was defending Trump's immigration crackdown.

During a 25-minute appearance on "The Lead with Jake Tapper" Tuesday, Conway criticized the media for sloppy reporting. But she conceded that it was unfair to label CNN as "fake news" -- as Trump has done previously.

"I don't think CNN is 'fake news.' I think there are some reports everywhere, in print, on TV, on radio, in conversation, that are not well-researched and are sometimes based on falsehoods," Conway said.

Trump has continually slammed major media organizations, including The New York Times and The Washington Post, as well as CNN, as "fake news."

Conway said she was "trying to reach out" to the media and "put out the olive branch."

At one point, Conway seemed to argue that the President offering blatant falsehoods as truth should matter less than his perceived accomplishments.

"Are they (falsehoods) more important than the many things that he says that are true that are making a difference in people's lives?" Conway asked.

Facts and falsehoods

Tapper asked Conway about some of Trump's false claims, including his claim earlier Tuesday that the murder rate was at the highest point in 47 years -- which is wrong

"Everyday there are these sprays of attack and sprays of falsehoods coming from the White House. It would be better if they were not coming from the White House, for me and for you," Tapper said, adding that it inhibits news organizations' ability to focus solely on policy.

"Agreed, and let me just say it has to go both ways. I do, Jake, I sincerely don't see a lot of difference in coverage from when he was a candidate and when he became the Republican nominee, the president-elect and, indeed the President," Conway said.

Conway also apologized for shaming the press for not covering the "Bowling Green Massacre," an incident which did not occur.

"I regretted it tremendously," Conway said. "I felt really badly about that."

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Terrorism coverage

Tapper asked Conway to grapple with the President's claim on Monday that the press does not cover terrorism and has an unstated reason for doing so. On Monday, the White House released a list of 78 attacks saying: "Most of these attacks did not receive adequate attention from Western media sources."

The list included many attacks covered extensively by CNN and other news outlets.

Conway conceded that many of the attacks were indeed covered extensively,

"Obviously, the very sad incidents that you've related ... frankly, CNN did amazing coverage for weeks at a time," Conway said.

But she said other attacks weren't covered enough relative to Trump's candidacy or the death of the singer Prince and alleged claims that former Secretary of State Hillary Clinton, Trump's 2016 rival, treated terrorism too lightly.

Tapper responded: "Facts are stubborn things, and to say that we're not reporting something that happens not to be true -- therefore we're not to be trusted -- that's a problem."

"I'm among, if not the most open press person in the White House ... I'm now being attacked by the media," Conway said. "I'm just going to keep soldiering on."

On the point of attacks Trump himself has ignored, Tapper asked about a recent shooting in Quebec, where a man stands accused of murdering Muslims. Trump did not issue a tweet or other public statement in response to the event.

"I know he's sympathetic to any loss of life," Conway said. "I will ask him. He doesn't tweet about everything."

Putin and policy

As Tapper pointed out, Trump's falsehoods and attacks on the press had dominated much of the coverage of the President, clouding out some potential policy coverage.

However, the two did manage to speak about governing at the beginning of the interview.

The White House adviser said she was glad the Senate had confirmed Betsy DeVos for education secretary earlier in the day.

While acknowledging she respected the concerns of the two dissenting Republican voices in the Senate, she praised DeVos and indicated the newly minted education secretary would help Trump follow through on his education promises.

"He wants to repeal Common Core," Conway said, adding that public education "doesn't work for everyone."

The interview also generated a response from Conway about Trump's most recent comments on Russia

In an interview that aired before the Super Bowl, Trump seemed to defend Russian President Vladimir Putin's human rights abuses, saying, "You think our country's so innocent?"

The comment generated an outpouring of criticism from people who said Trump was brushing off crimes like occupying the territory of other countries, violently crushing dissent and propping up despots.

"A lot of this stems from the fact that there just seems to be charge and accusation after charge and accusation that somehow President Trump and Vladimir Putin are BFFs. That is not true," Conway said.

Conway said she did not think Trump was drawing a moral equivalence between the US and Russia, adding, "He was thinking about the war in Iraq, and in that regard, I think people should make the judgment for themselves."