Product structure of graphs excluding a topological minor

We prove that, for all positive integers $h$ and $t$ and every graph $X$ with $\mathrm{t}\mathrm{d}(X)\le h$, there exists a positive integer $c(X,t)$ such that every graph $G$ with $\mathrm{t}\mathrm{w}(G)<t$ that excludes $X$ as a topological minor is isomorphic to a subgraph of $H⊠K_{c(X,t)}$ for some graph $H$ with $\mathrm{t}\mathrm{w}(H)<2^{h+1}-1$. This extends a result by Ding and Oporowski (Journal of Graph Theory; 1995), which states that for all positive integers $\mathrm{\Delta }$ and $t$, here exists a positive integer $f(\mathrm{\Delta },t)$ such that every graph $G$ with $\mathrm{t}\mathrm{w}(G)<t$ and $\mathrm{\Delta }(G)\le \mathrm{\Delta }$ is isomorphic to a subgraph of $T⊠K_{f(\mathrm{\Delta },t)}$ for some tree $T$.