When does a bipartite graph have bounded treewidth?
I think there is no simpler characterization than just the fact that the treewidth of $G$ is bounded. The intuition for why is that by subdividing each edge of a graph we get a bipartite graph with the same treewidth. In particular, we have a very simple reduction from determining the treewidth of a graph to determining the treewidth of a bipartite graph.