Turán number for families of graphs behave like Turán number for one of the members

For a finite family \( {\mathcal F}\) of graphs, let \( t(n, {\mathcal{F}})\) denote the smallest integer \( m\) that every graph on \( n\) vertices and \( m\) edges must contain a member of \( \mathcal{F}\) as a subgraph. Erdös and Simonovits [1] raised the following question:

A problem on the Turán number for families of graphs}

Prove that \[ t(n,{\mathcal F}) = O(t(n,G)) \] for some graph \(G\) in \(\mathcal{F}\).

1 P. Erdös and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), 275-288.