Steady-state two-dimensional lid-driven square cavity flows at high Reynolds numbers are solved in this communication using a velocity-based formulation developed in the context of the improved element-free Galerkin–reduced integration penalty method (IEFG–RIPM). The analyses based on the IEFG–RIPM are performed under a standard Galerkin weak-formulation, i.e. without the need of introducing streamline-upwind or pressure stabilizing terms in order to suppress the appearance of non-physical oscillations. Solutions in a wide range of high Reynolds numbers are successfully achieved, and the results obtained have exhibited an excellent agreement with mesh-based solutions reported in the literature. The numerical performance of the proposed IEFG–RIPM in the solution of such benchmark problem has been analysed in terms of velocity distribution, streamline patterns, vorticity and pressure contours, and also in terms of the properties of primary and secondary vortices. The outcomes of this study demonstrate the potential of the IEFG–RIPM as a feasible and reliable numerical technique for the analysis of lid-driven square cavity flows at high Reynolds numbers, allowing the fulfilment of accuracy and stability numerical requirements demanded in the solution of this benchmark problem in a remarkably simple manner.