Journalists and commentators, meanwhile, have said a lot about Mr. Bush over the last few months. Article after article has depicted him as a relative moderate, as a candidate of the establishment seeking to consolidate the support of the party’s elite. It’s not hard to imagine why this hasn’t worn well among the party’s conservative, populist base, and why his ratings among conservatives may have sunk as a result.

His challenges probably run beyond specific policy grievances on Common Core or immigration. Marco Rubio, for instance, holds policy views similar to those of Mr. Bush, including his own weakness among the base on immigration reform. But 74 percent of voters in Iowa with an opinion felt Mr. Rubio was “about right” ideologically, compared with just 48 percent who thought the same thing about Mr. Bush.

This is a serious challenge to Mr. Bush, but it is not insurmountable. He doesn’t need to win the very conservative or evangelical vote to win the party’s nomination, even if he does need to be sufficiently strong among those groups to prevent an opponent from building a large enough coalition to win. Scott Walker, an evangelical Christian with a record of taking on labor unions in Wisconsin, is well positioned to do so. The same Iowa poll gave Mr. Walker a nominal lead over the field; perhaps more important, 85 percent of voters with an opinion of Mr. Walker thought he was ideologically “about right.”