In order to analyze the nonlinear propagation of arbitrary amplitude ion-acoustic waves, the Sagdeev pseudopotential formalism is used to a four-component multi-ion plasma made up of inertial light positive ions, heavy negative ions, inertialess nonextensive electrons, and positrons. At first, we derived the expression of Sagdeev Pseudopotential for our plasma model and then numerically investigated the condition for the existence of several nonlinear structures such as solitons, double layers, supersolitons, etc. Based on numerical simulation, it is observed that within a certain range of parameters, our plasma model allows compressive as well as a rarefactive soliton, flat-top soliton, double layer, and supersoliton solutions. Great emphasis is placed on the determination of the existence regions for both solitons and supersolitons in the parameter space. Also, various nonlinear structures are explored to examine the effects of various plasma properties, such as the nonextensivity parameter, Mach number, etc. The findings of the current investigations may be helpful for comprehending significant features of ion-acoustic wave structures in space plasmas such as earth's ionosphere, cometary tails, etc., and laboratory plasmas like laser-plasma interaction research.