When former Alaska Gov. Sarah Palin signed a contract in January to become a regular guest on Fox News, many of her critics said it was part of a strategy by the Republican to avoid tough or skeptical questions and engage only with friendly members of the media. But Bill O’Reilly appears to have missed that memo. The larger-than-life Fox News host with the top rated cable news show on TV has a history of challenging Palin. “You don’t have that much experience,” he told the former governor last fall, though he has also defended her performance as governor. But...