<p>This paper introduces a relaxed iterative method for approximating solutions to bilevel variational inequality problems (BVIP) in real Hilbert spaces framework. The mathematical structure addressed herein involves two variational inequalities: one constrained by the other, where the mapping in the lower level is pseudomonotone uniformly continuous. We propose an iterative algorithm that specifically addresses cases where the lower-level mapping is pseudomonotone and uniformly continuous. We then establish strong convergence of the generated sequence to a unique solution of the BVIP using an Armijo-line search procedure. We also provide numerical illustrations in infinite-dimensional Hilbert space settings, including a comparative analysis with previous methods, demonstrating that the proposed approach achieves faster convergence and requires less computational time. A significant contribution of this work is the removal of restrictive assumptions commonly found in existing literature, such as weak sequential continuity and other specific conditions for the lower-level mapping. Furthermore, the method improves computational efficiency by reducing the number of metric projections required to just one per iteration.</p>