The purpose of this manuscript was to introduce a new iterative approach based on Green's function for approximating numerical solutions of the Troesch's problem. A Banach space of continuous functions was considered for establishing the main outcome. First, we set an integral operator using a Green's function and embedded this new operator into a three-step iterative scheme. We proved the main convergence result with the help of some mild assumptions on the parameters involved in our scheme and in the problem. Moreover, we proved that the new iterative approach was weak $w^2$-stable. The high accuracy and stability of the scheme was confirmed by several numerical simulations. As an application of the main result, we solved a class of fractional boundary value problems (BVPs). The main results improved and unified several known results of the literature.