Extremal graph avoiding \(3\)- and \(4\)-cycles avoids all odd cycles

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. Simonovits [1] raised the following question:

Conjecture[1]

Prove that \[ t(n,\{C_{3},C_{4}\}) = t(n,\{C_{3},C_{4},C_5,C_7,C_9,C_{11},\ldots\}). \]

Bibliography
1 M. Simonovits, Paul Erdös' influence on extremal graph theory, The Mathematics of Paul Erdös, II (R. L. Graham and J. Nešetril, eds.), 148-192, Springer-Verlag, Berlin, 1996.