  Given two graphs $G, H$ and a positive integer $q$, an $(H,q)$-coloring of
$G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ receives at
least $q$ distinct colors. The bipartite Erd\H{o}s-Gy\'{a}rf\'{a}s function
$r(K_{n,n}, K_{s,t}, q)$ is defined to be the minimum number of colors needed
for $K_{n,n}$ to have a $(K_{s,t}, q)$-coloring. For balanced complete
bipartite graphs $K_{p,p}$, the function $r(K_{n,n}, K_{p,p}, q)$ was studied
systematically in [Axenovich, F\"{u}redi and Mubayi, {\it J. Combin. Theory
Ser. B} {\bf 79} (2000), 66--86]. In this paper, we study the asymptotic
behavior of this function for complete bipartite graphs $K_{s,t}$ that are not
necessarily balanced. Our main results deal with thresholds and lower and upper
bounds for the growth rate of this function, in particular for (sub)linear and
(sub)quadratic growth. We also obtain new lower bounds for the balanced
bipartite case, and improve several results given by Axenovich, F\"{u}redi and
Mubayi. Our proof techniques are based on an extension to bipartite graphs of
the recently developed Color Energy Method by Pohoata and Sheffer and its
refinements, and a generalization of an old result due to Corr\'{a}di.
