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.
Modern liberalism in the United States
In the first half of the twentieth century, both major American parties had a conservative wing and a liberal wing. The conservative northern Republicans and the conservative southern Democrats formed the Conservative Coalition which dominated the US congress in the pre-Civil Rights era. As the Democratic Party under President Lyndon Johnson began to support civil rights, the formerly Solid South, meaning solidly Democratic, became solidly Republican, except in districts with a large number of African-American voters.