We introduce the iterative Clifford circuit renormalization (ICCR), a novel technique designed to handle the dynamics of nonstabilizerness (also known as quantum magic) in generic quantum circuits composed of Clifford and non-Clifford gates, as well as measurements, extending beyond the reach of pre-existing methods. ICCR iteratively adjusts the starting circuit, transforming it into a Clifford circuit where all elements that can alter the nonstabilizerness, such as measurements or T gates, have been removed. In the process the initial state is renormalized in such a way that the new circuit outputs the same final state as the original one. This approach embeds the complex dynamics of nonstabilizerness in the flow of an effective initial state, enabling the efficient evaluation of stabilizer Renyi entropies and magic nullity while avoiding the need for direct and computationally expensive simulation of the original circuit. The initial state renormalization can be computed explicitly using a matrix-product state approximation that can be systematically improved. We implement the ICCR algorithm to evaluate the nonstabilizerness dynamics for systems of size up to N = 1000, in one and more dimensions. We validate our method by comparing it to tensor-network simulations. Finally, we employ the ICCR technique to study measurement-induced magic transitions in monitored circuits without and with T gates.
