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.
Conservative Party (UK)
The second main grouping in the Conservative Party is the "free-market wing" of economic liberals who achieved dominance after the election of Margaret Thatcher as party leader in 1975. Their goal was to reduce the role of the government in the economy and to this end they supported cuts in direct taxation, the privatisation of nationalised industries and a reduction in the size and scope of the welfare state. Supporters of the "free-market wing" have been labelled as "Thatcherites". The group has disparate views of social policy: Thatcher herself was socially conservative and a practising Anglican but the free-market wing in the Conservative Party harbour a range of social opinions from the civil libertarian views of Michael Portillo, Daniel Hannan, and David Davis to the traditional conservatism of former party leaders William Hague and Iain Duncan Smith. The Thatcherite wing is also associated with the concept of a "classless society".