Let {mathcal {C}} and {mathcal {D}} be hereditary graph classes. Consider the following problem: given a graph Gin {mathcal {D}}, find a largest, in terms of the number of vertices, induced subgraph of G that belongs to {mathcal {C}}. We prove that it can be solved in 2^{o(n)} time, where n is the number of vertices of G, if the following conditions are satisfied:the graphs in {mathcal {C}} are sparse, i.e., they have linearly many edges in terms of the number of vertices;the graphs in {mathcal {D}} admit balanced separators of size governed by their density, e.g., {mathcal {O}}(varDelta ) or {mathcal {O}}(sqrt{m}), where varDelta and m denote the maximum degree and the number of edges, respectively; andthe considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes {mathcal {C}} and {mathcal {D}}:a largest induced forest in a P_t-free graph can be found in 2^{tilde{{mathcal {O}}}(n^{2/3})} time, for every fixed t; anda largest induced planar graph in a string graph can be found in 2^{tilde{{mathcal {O}}}(n^{2/3})} time.