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.
Conservatism
The meaning of "conservatism" in the United States has little in common with the way the word is used elsewhere. As Ribuffo (2011) notes, "what Americans now call conservatism much of the world calls liberalism or neoliberalism". Since the 1950s, conservatism in the United States has been chiefly associated with the Republican Party. However, during the era of segregation many Southern Democrats were conservatives and they played a key role in the Conservative coalition that largely controlled domestic policy in Congress from 1937 to 1963, The Conservative Democrats continued to have influence in the U.S. politics until 1994's Republican Revolution, when the American South shifted from solid Democrat to solid Republican, while maintaining its conservative values.