Behavior of Turán number for a family of graphs is controlled by a bipartite member
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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), 275288.
