The principle aim of this manuscript is to propose a general scheme that can be applied to any optimal iteration function of order eight whose first substep employ Newton’s method to further develop new interesting optimal scheme of order sixteen. This scheme requires four evaluations of the involved function and one evaluation of its first-order derivative at each step. So, it is being optimally consistent with the conjecture of Kung–Traub. In addition, theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence. Moreover, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the simple roots. From the numerical experiments, we find that our methods perform better than the existing ones when we checked the performance on a concrete variety of nonlinear equations.