If $\cal {G}$ is a finitely generated group with generators $\{g_1,\dots,g_j\}$ , then an infinite-order element $f \in \cal {G}$ is a distortion element of $\cal {G}$ provided that ${\lim \, \inf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface, and let $\rm{Diff}(S)_0$ denote the identity component of the group of $C^1$ -diffeomorphisms of $S$ . Our main result shows that if $S$ has genus at least two and that if $f$ is a distortion element in some finitely generated subgroup of $\rm{Diff}(S)_0$ , then $\rm {supp}(\mu) \subset \rm {Fix}(f)$ for every $f$ -invariant Borel probability measure $\mu$ . Related results are proved for $S = T^2$ or $S^2$ . For $\mu$ a Borel probability measure on $S$ , denote the group of $C^1$ -diffeomorphisms that preserve $\mu$ by $\rm{Diff}_{\mu}(S)$ . We give several applications of our main result, showing that certain groups, including a large class of higher-rank lattices, admit no homomorphisms to $\rm{Diff}_{\mu}(S)$ with infinite image