The present study employs time-resolved two-dimensional numerical simulations to investigate the complex flow dynamics of viscoelastic fluids within a two-sided lid-driven square cavity. In this configuration, the upper wall moves toward the right, while the left wall moves downward at identical velocities. Viscoelastic fluids \(\left( {{\text{VE}}} \right)\) , attained by introducing minute polymer compounds into Newtonian fluids such as water, exhibit distinctive flow behavior. These fluids can trigger elastic instabilities even at exceedingly low Reynolds numbers. Subsequently, these instabilities transition toward the regime of elastic turbulence \(\left( {{\text{ET}}} \right)\) as the Weissenberg number \(\left( {{\text{Wi}}} \right)\) experiences a gradual increase. In the present study, we conducted numerical simulations at Reynolds number \(\left( {{\text{Re}} = 0.0001} \right)\) to examine the influence of purely elastic instabilities on flow bifurcation within the context of the two-sided lid-driven cavity. This study employs the Oldroyd-B constitutive \({\text{VE}}\) model to understand the repercussions of purely elastic instabilities comprehensively. This model replicates \({\text{VE}}\) fluids’ rheological attributes with a constant shear viscosity, commonly called Boger fluids. The numerical computations are carried out using the finite volume-based solver rheoFOAM, an integral component of the rheoTOOL package. In the case of Newtonian fluids, the resulting flow pattern remains consistently steady and symmetric about one of the diagonals of the cavity. However, when the \({\text{Wi}}\) surpasses the critical threshold of \(0.8\) , a transition emerges: the flow evolves into the \({\text{ET}}\) regime. This transition is marked by significant deviations in the flow bifurcation parameter from its mean value. Furthermore, a closer examination of the time-averaged streamline plots unveils a compelling phenomenon: as the \({\text{Wi}}\) progressively increases, the vortex centers of the right primary vortex (RPV) and left primary vortex (LPV) migrate toward corner A (see Fig. 1). It’s worth noting that the outcomes of this study offer a valuable reference for validating the intricate flow patterns of \({\text{VE}}\) fluids. These patterns arise from the complex interplay between two vortices within a two-sided lid-driven square cavity.