The extremal function for bipartite linklessly embeddable graphs

An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple) graph on $n\ge 5$ vertices has at most $3n-10$ edges, unless it is isomorphic to the complete bipartite graph $K_{3,n-3}$.