  Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge
ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for
bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is
the graph associated with the polarization of the ideal $(I(G)^{s+1} :
e_1\cdots e_s)$, where $e_1,\ldots e_s$ are edges of $G$. Using these results,
we explicitly compute reg$(I(G)^s)$ for several subclasses of bipartite graphs.
