Finding subgraphs with high treewidth and constant degree
See the paper by Julia Chuzhoy and myself on Treewidth sparsifiers. 
We show that one can obtain a subgraph of degree at most 3 with treewidth $\Omega(k/polylog(k))$ where $k$ is the treewidth of $G$. 
https://arxiv.org/abs/1410.1016 The proof is shorter than the one 
for grid minors but it is still not that that easy and builds on several previous 
tools.

Suppose you settle for an easier target - degree 4 and treewidth $\Omega(k^{1/4})$ then you can get it much more easily via  result of Reed and Wood on grid-like minors. https://arxiv.org/abs/0809.0724

Another easy result you can obtain is the following which is a starting point for some of the more involved proofs. You can get a subgraph of degre  $\log^2(k)$ 
and treewidth $\Omega(k/\mathsf{polylog}(k))$. You can see the treewidth sparsifier paper for the argument to achieve this.