Relation between tree-width and clique number
On this page a theorem is mentioned that provides such classes:

Theorem (Scheffler [1]) If $G$ is the intersection graph of connected subgraphs of another graph $H$, then $tw(G)\leq tw(H)\omega(G)-1$.

This generalizes the bound for chordal graphs (for which $H$ is a tree) and also applies to circular-arc graphs (then $H$ is a cycle). I do not know if other "standard" classes are captured by this theorem.



[1] P. Scheffler, What graphs have bounded tree-width? Rostocker Math. Kolloq. 41 (1990) 31-38.