  A graph is locally irregular if no two adjacent vertices have the same
degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is
the smallest number of locally irregular subgraphs needed to edge-decompose
$G$. Not all graphs have such a decomposition, but Baudon, Bensmail,
Przyby{\l}o, and Wo\'zniak conjectured that if $G$ can be decomposed into
locally irregular subgraphs, then $\chi_{\rm irr}'(G)\leq 3$. In support of
this conjecture, Przyby{\l}o showed that $\chi_{\rm irr}'(G)\leq 3$ holds
whenever $G$ has minimum degree at least $10^{10}$.
  Here we prove that every bipartite graph $G$ which is not an odd length path
satisfies $\chi_{\rm irr}'(G)\leq 10$. This is the first general constant upper
bound on the irregular chromatic index of bipartite graphs. Combining this
result with Przyby{\l}o's result, we show that $\chi_{\rm irr}'(G) \leq 328$
for every graph $G$ which admits a decomposition into locally irregular
subgraphs. Finally, we show that $\chi_{\rm irr}'(G)\leq 2$ for every
$16$-edge-connected bipartite graph $G$.
