An operator $T$ is called $n$-normal operator if $T^nT^* = T^*T^n$ and $n$-quasinormal operator if $T^nT^*T = T^*TT^n$. In this paper, the conditions under which composition operators and weighted composition operators become $n$-normal operators and $n$-quasinormal operators have been obtained in terms of Radon-Nikodym derivative $h_n$.