The paper introduces the Iterative Clifford Circuit Renormalization (ICCR) technique, designed to efficiently handle the dynamics of non-stabilizerness (quantum magic) in generic quantum circuits. ICCR iteratively adjusts the starting circuit to transform it into a Clifford circuit, removing elements that can alter non-stabilizerness, such as measurements or $T$ gates. The initial state is renormalized to output the same final state as the original circuit, enabling efficient evaluation of non-stabilizerness dynamics without direct simulation. The method is applied to systems up to $N = 1000$ qubits, validated against tensor network simulations, and used to study a magic purification circuit, observing a measurement-induced transition in the relaxation dynamics of non-stabilizerness measures. The ICCR algorithm is detailed, including its implementation and computational cost, and its performance is discussed, showing good agreement with exact simulations. The technique is versatile and can be applied to various problems involving non-stabilizerness dynamics, particularly in large systems and high-dimensional or long-range circuits.The paper introduces the Iterative Clifford Circuit Renormalization (ICCR) technique, designed to efficiently handle the dynamics of non-stabilizerness (quantum magic) in generic quantum circuits. ICCR iteratively adjusts the starting circuit to transform it into a Clifford circuit, removing elements that can alter non-stabilizerness, such as measurements or $T$ gates. The initial state is renormalized to output the same final state as the original circuit, enabling efficient evaluation of non-stabilizerness dynamics without direct simulation. The method is applied to systems up to $N = 1000$ qubits, validated against tensor network simulations, and used to study a magic purification circuit, observing a measurement-induced transition in the relaxation dynamics of non-stabilizerness measures. The ICCR algorithm is detailed, including its implementation and computational cost, and its performance is discussed, showing good agreement with exact simulations. The technique is versatile and can be applied to various problems involving non-stabilizerness dynamics, particularly in large systems and high-dimensional or long-range circuits.