What is the treewidth of $K_{m,n}$?
Let $l=\min\{m,n\}$ and $k=\lfloor \sqrt{2l}\rfloor$. A $k$ by $k$ grid is bipartite and contains $\lfloor k^2/2 \rfloor$ vertices in the smaller part and $\lceil k^2/2\rceil$ vertices in the larger part. Hence $K_{m,n}$ contains a $k$ by $k$ grid as a subgraph, so its treewidth is at least $k = \lfloor \sqrt{2\cdot \min\{m,n\}}\rfloor$.
Is this tight, or if not, is there a more precise expression?
A graph of treewidth $k$ must be $k$-degenerate. Since $K_{m,n}$ has degeneracy $l=min(m,n)$, the treewidth is at least $l$.
It is at most $l$: let $S$ be the set with $l$ vertices and the other vertices be $a_1,a_2,\ldots,a_n$ (say). The tree is $S$ as root and $S \cup \{a_i\}$ as children.