Varieties of criticism
Conservative criticism has nothing much to do with "left-wing" or "right-wing", because left-wing people are often very conservative, in the defined sense, while right-wing people can also be very radical, in the defined sense. The difference between "radical" and "conservative" has more to do with the belief in whether "a change to something genuinely new" is really possible and necessary. Radicals typically believe strongly that such change is highly desirable and necessary, and that it can be achieved. Their criticism is that there is "not enough" change. Conservative critics, by contrast, are very skeptical about any such change, because they feel the change is really "just another form" of something that already exists.
Factions in the Democratic Party (United States)
The conservative coalition was an unofficial coalition in the United States Congress bringing together a conservative majority of the Republican Party and the conservative, mostly Southern wing of the Democratic Party. It was dominant in Congress from 1937-1963 and remained a political force until the mid-1980s, eventually dying out in the 1990s. In terms of Congressional roll call votes, it primarily appeared on votes affecting labor unions. The conservative coalition did not operate on civil rights bills, for the two wings had opposing viewpoints. However, the coalition did have the power to prevent unwanted bills from even coming to a vote. The coalition included many committee chairmen from the South who blocked bills by not reporting them from their committees. Furthermore, Howard W. Smith, Chairman of the House Rules Committee, often could kill a bill simply by not reporting it out with a favorable rule and he lost some of that power in 1961. The conservative coalition was not concerned with foreign policy as most of the Southern Democrats were internationalists, a position opposed by most Republicans.