Excluding subdivisions of bounded degree graphs

Let $H$ be a fixed graph. What can be said about graphs $G$ that have no subgraph isomorphic to a subdivision of $H$? Grohe and Marx proved that such graphs $G$ satisfy a certain structure theorem that is not satisfied by graphs that contain a subdivision of a (larger) graph $H_1$. Dvo\v{r}\'ak found a clever strengthening---his structure is not satisfied by graphs that contain a subdivision of a graph $H_2$, where $H_2$ has "similar embedding properties" as $H$. Building upon Dvo\v{r}\'ak's theorem, we prove that said graphs $G$ satisfy a similar structure theorem. Our structure is not satisfied by graphs that contain a subdivision of a graph $H_3$ that has similar embedding properties as $H$ and has the same maximum degree as $H$. This will be important in a forthcoming application to well-quasi-ordering.