It's a major open question to determine this value, even up to constant factors, even when $n=m$. In this special case, the best upper bound has the form $|E| = O(n^{1 + 1/4})$ (see this paper or many others), while the best lower bound has the form $|E| = \Omega(n^{1 + 1/5})$ (the only lower bound graph I'm aware of is the incidence graph of the Split Cayley Hexagon, but unfortunately I can't point you at a good reference for how to build this, and I don't know if there's a simpler construction out there. Maybe someone can help me out here).

For the general case when $n < m$, it should be easy to extend the results of the above upper bound paper to prove $|E| = O(n^{3/4} m^{1/2} + m)$ (I did some back-of-the-envelope computations here but I haven't totally verified this, so beware). The only interesting lower bound graph I'm aware of is the incidence graph of the twisted triality hexagon, which I believe requires $m = n^3$ and then gives $|E| = \Omega(n^{20/7})$.

It's a major open question to determine this value, even up to constant factors, even when $n=m$. In this special case, the best upper bound has the form $|E| = O(n^{1 + 1/4})$ (see this paper or many others), while the best lower bound has the form $|E| = \Omega(n^{1 + 1/5})$ (the only lower bound graph I'm aware of is the incidence graph of the Split Cayley Hexagon, but unfortunately I can't point you at a good reference for how to build this, and I don't know if there's a simpler construction out there. Maybe someone can help me out here).

For the general case when $n < m$, it should be easy to extend the results of the above upper bound paper to prove $|E| = O(n^{3/4} m^{1/2} + m)$ (I did some back-of-the-envelope computations here but I haven't totally verified this, so beware). The only interesting lower bound graph I'm aware of is the incidence graph of the twisted triality hexagon, which I believe requires $m = n^{5/4}$ and then gives $|E| = \Omega(n^{11/8})$.

It's a major open question to determine this value, even up to constant factors, even when $n=m$. In this special case, the best upper bound has the form $|E| = O(n^{1 + 1/4})$ (see this paper or many others), while the best lower bound has the form $|E| = \Omega(n^{1 + 1/5})$ (the only lower bound graph I'm aware of is the incidence graph of the Split Cayley Hexagon, but unfortunately I can't point you at a good reference for how to build this, and I don't know if there's a simpler construction out there. Maybe someone can help me out here).

For the general case when $n < m$, it should be easy to extend the results of the above upper bound paper to prove $|E| = O(n^{3/4} m^{1/2} + m)$ (I did some back-of-the-envelope computations here but I haven't totally verified this, so beware). I don't know of any interesting lower bound constructions.

It's a major open question to determine this value, even up to constant factors, even when $n=m$. In this special case, the best upper bound has the form $|E| = O(n^{1 + 1/4})$ (see this paper or many others), while the best lower bound has the form $|E| = \Omega(n^{1 + 1/5})$ (the only lower bound graph I'm aware of is the incidence graph of the Split Cayley Hexagon, but unfortunately I can't point you at a good reference for how to build this, and I don't know if there's a simpler construction out there. Maybe someone can help me out here).

For the general case when $n < m$, it should be easy to extend the results of the above upper bound paper to prove $|E| = O(n^{3/4} m^{1/2} + m)$ (I did some back-of-the-envelope computations here but I haven't totally verified this, so beware). The only interesting lower bound graph I'm aware of is the incidence graph of the twisted triality hexagon, which I believe requires $m = n^3$ and then gives $|E| = \Omega(n^{20/7})$.