Until Megyn Kelly, no prime-time Fox News anchor had tried to leap from partisan basic cable to the more pedigreed world of network news.

Less than a month into her tenure at NBC, Ms. Kelly and her new employer — which has placed a multimillion-dollar bet on her success — are learning just how daunting the transition can be.

Even before it airs on Sunday, Ms. Kelly’s interview with Alex Jones, the conspiracy-monger and influential voice of the so-called alt-right, a far-right, white nationalist movement, has generated a fierce backlash, just as the anchor is introducing herself to a broader audience.

Parents of children killed in the 2012 Sandy Hook elementary school shooting, which Mr. Jones called a hoax, asked NBC to spike the interview, saying it was extremely hurtful for her to offer a platform for Mr. Jones’s views. Ms. Kelly was disinvited from a Sandy Hook charity event and accused by some viewers of chasing ratings by infecting NBC with Fox News-style conservatism.