Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $Δ$, then the choosability (or list chromatic number) of $G$ satisfies $χ_{\ell}(G) = O \left ( \log Δ\right )$. Currently, the best known upper bound for $χ_{\ell}(G)$ is $(1 + o(1)) \fracΔ{\log Δ}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $Δ$ satisfies $χ_{\ell}(G) < (\frac{4}{5} -\varepsilon) \fracΔ{\log Δ}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich.