We prove that for every bipartite graph H and positive integer s, the class of Kₒ, ₒ-subgraph-free graphs excluding H as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every Kₒ, ₓ-free bipartite circle graph with s t has a vertex of degree at most \2s-2, t-1\ and provide examples showing that this is tight.
Campbell et al. (Thu,) studied this question.