The Erd\H{o}s-Rogers function $fs,t$ measures how large a $Ks$-free induced subgraph there must be in a $Kt$-free graph on $n$ vertices. While good estimates for $fs,t$ are known for some pairs $(s,t)$, notably when $t=s+1$, in general there are significant gaps between the best known upper and lower bounds. We improve the upper bounds when $s+2\le t\le 2s-1$. For each such pair we obtain for the first time a proof that $fs,t\le n\alpha s,t+o(1)$ with an exponent , answering a question of Dudek, Retter and R"{o}dl.