In the present study, fourth-order compact formulation has been improved for Navier-Stokes (N-S) equations, which is expressed for two-dimensional, steady, incompressible flow problems. N-S equation system has been expressed with Stream Function-Vorticity Approach using Finite Difference Method (FDM) from the numerical methods. In order to test the functionality and applicability of the improved numerical formulation, a sample for submerged bluff bodies, flow problem around cylinder with square cross-section was chosen as a benchmark problem. As a result of applying improved numerical formulation with Gauss-Seidel Relaxation Method was used for this benchmark problem. The benchmark problem was also solved with second-order accuracy and obtained numerical results were compared with fourth-order accuracy numerical results. With the same Reynolds Number and the same free-stream velocity values, fourth-order numerical results are more convergent than second-order numerical results. Furthermore, in the flow field for considered benchmark problem, separation bubble length that consisted in the wake region is increased proportionally depending on the alteration of the Reynolds Number values.