The general theory of representations of the Cartan type infinite-dimensional Lie algebras was initiated by A. N. Rudakov. He studied topological irreducible representations of these algebras [Math. USSR, Izv. 8, 836-866 (1974); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 38, 835-866 (1974; Zbl 0322.17004) and Math. USSR, Izv. 9 (1975), 465-480 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 39, 496-511 (1975; Zbl 0345.17008)]. In this paper, the authors study polynomial Cartan type Lie algebras \(W_n\) and \(\overline S_n\) and, without imposing any continuity condition, investigate arbitrary irreducible weight representations. The main result is an explicit description of all possible sets of weights of such representations, i.e. a description of the supports of all irreducible weight representations of \(W_n\) and \(\overline S_n\). The key technical feature that made this description possible is that any parabolic subalgebra of \(\text{sl}_{n+1}\) has a certain canonical extension to a parabolic subalgebra of \(W_n\) or \(\overline S_n\).