  Previously, Ohsugi and Hibi gave a combinatorial description of bipartite
graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that
every cycle of $G$ of length at least $6$ must have a chord. This corresponds
to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate
the higher syzygies of $I_G$ and give combinatorial descriptions of the
Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite
graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly
presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if the
bipartite complement of $G$ is a tree of diameter at most $3$. We also
investigate the regularity of linearly presented toric edge ideals and give
criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions.
