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)
Today, conservative Democrats are generally regarded as members of the Democratic Party who are more conservative than the national political party as a whole. Conservative Democrats vary greatly in ideology. Some are fiscally conservative and socially liberal Democrats while some are fiscally conservative and socially moderate Democrats. On foreign policy, conservative Democrats are generally liberal internationalists.