P4-sparse graphs (defined by Hoàng) and P4-reducible graphs (defined by Jamison and Olariu) are graphs with weak density of P4's. Weak-bisplit graphs are bipartite graphs which show some analogies with cographs (i.e. P4-free graphs) and are characterized with two forbidden configurations. We describe here bipartite graphs with weak density of those configurations. Structural properties and recognition algorithms are given.