A bipartite graph G=(A,B,E) is H-convex for some family of graphs H if there exists a graph H∈H with V(H)=A such that the neighbours in A of each b∈B induce a connected subgraph of H. Many NP-complete problems are polynomial-time solvable for H-convex graphs when H is the set of paths. The underlying reason is that the class has bounded mim-width. We extend this result to families of H-convex graphs where H is the set of cycles, or H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we strengthen many known results via one general and short proof. We also show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3.