Given a graph parameter P, we say that a graph family F has the parameter-treewidth property for P if there is a function f(p) such that every graph G in F with parameter at most p has treewidth at most f(p). We prove as our main result that, for a large family of parameters called contraction-bidimensional parameters, a minor-closed graph family F has the parameter-treewidth property if F has bounded local treewidth. We also show “if and only if” for some parameters, and thus this result is in some sense tight. In addition we show that, for a slightly smaller family of parameters called minor-bidimensional parameters, all minor-closed graph families F excluding some fixed graphs have the parameter-treewidth property. The bidimensional parameters include many domination and covering parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, q-dominating set (for fixed q). We use these theorems to develop new fixed-parameter algorithms in these contexts.