The focus of this paper is to introduce a four step iterative algorithm, called A* iterative method, for approximating the fixed points of Suzuki generalized nonexpansive mappings. We prove analytically and numerically that our new iterative algorithm converges faster than some leading iterative algorithm in the literature for almost contraction mappings and Suzuki generalized nonexapansive mapping. Furthermore, we prove weak and strong convergence theorems of our new iterative method for Suzuki generalized nonexpansive mappings in uniformly convex Banach spaces. Again, we show analytically and numerically that our new iterative algorithm is G-stable and data dependent. Finally, to illustrate the applicability of our new iterative method, we will find the unique solution of a functional Volterra Fredholm integral equation with a deviating argument via our new iterative method. Hence, our results generalize and improve several well known results in the existing literature.