Behavior of Turán number for a family of graphs is controlled by a bipartite member

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.

Conjecture[1]

For every family \(\mathcal F\) of graphs containing a bipartite graph, there is a bipartite graph \(B \in {\mathcal F}\) for which \[ t(n, {\mathcal F})= O(t(n,B)). \]

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