  We study the Excluded Grid Theorem of Robertson and Seymour. This is a
fundamental result in graph theory, that states that there is some function $f:
Z^+\rightarrow Z^+$, such that for all integers $g>0$, every graph of treewidth
at least $f(g)$ contains the $(g\times g)$-grid as a minor. Until recently, the
best known upper bounds on $f$ were super-exponential in $g$. A recent work of
Chekuri and Chuzhoy provided the first polynomial bound, by showing that
treewidth $f(g)=O(g^{98}\operatorname{poly}\log g)$ is sufficient to ensure the
existence of the $(g\times g)$-grid minor in any graph. In this paper we
improve this bound to $f(g)=O(g^{19}\operatorname{poly}\log g)$. We introduce a
number of new techniques, including a conceptually simple and almost entirely
self-contained proof of the theorem that achieves a polynomial bound on $f(g)$.
