Asymptotic behavior for joint Turán number for \(3\)- and \(4\)-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. Erdös and Simonovits [1] raised the following question:

A problem on the Turán number of \(C_3\) and \(C_4\)[1]

Is it true that \[ t(n,\{C_3,C_4\}) = \frac 1 {2 \sqrt 2} n^{3/2} + O(n)~ ?\]

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