From a glitzy new office in downtown Washington, the ideological war over the media is fully engaged. Six years after its founding to counter what it said was “conservative misinformation,” Media Matters for America employs a staff of 70 that spends 19 hours a day monitoring newspapers, magazines, broadcast and cable television, talk radio, and the Internet to counter reporting or commentary it deems to be inaccurate or biased. Lou Dobbs recently described the group as part of “the vast left wing conspiracy,” an ironic twist on Hillary Clinton’s famous description of the conservative infrastructure arrayed against her husband when...