In the framework of dynamical Chern–Simons theory of gravity, we study the recent cosmic expansion with acceleration of the universe. We take interacting scenario of dark energy and dark matter with three cutoffs like Granda–Oliveros, higher order derivative of Hubble parameter and generalized holographic dark energy model. In the flat Friedmann–Robertson–Walker universe, well-known cosmological parameters have been calculated. We study the perturbation of matter density growth, growth factor and growth index behavior for the underlying framework. The cosmological parameters like equation of state parameter, deceleration parameter, and stability of each model are discussed. The outcomes of these models represent the cosmic expansion of the universe with acceleration.