Adding something further: the following is a theorem of Simonovits, which more or less everyone seems to have forgotten exists (buried in some conference proceedings, I think from the mid-80s) that more or less reduces the general problem to the (very interesting) bipartite case.

Given any graph $H$, the decomposition family of $H$ is the collection of bipartite graphs which arise as subgraphs of $H$ induced by any two colour classes in any proper $\chi(H)$-colouring of $H$.

Let the extremal number of the decomposition family be $f_H(n)$. If $f_H(n)$ grows superlinearly, then the difference between the extremal number of $H$ and of $K_{\chi(H)}$ is $\Theta(f_H(n))$.

In fact, it's not so hard to extend this to allow $f_H(n)$ to grow linearly; Simonovits didn't do this (or at least, if he did it is even more buried), and as far as I know this is not in the literature.

In any case, this gives that the extremal number of $K_{1,t,t}$ is $n^2/2+O(n)$. It's maybe a good exercise to follow LouisD's suggestion to prove the exact value; it works with no particular difficulty following the method in David Conlon's notes. The extremal graphs are obtained from the complete balanced bipartite graph on $n$ vertices by adding edge-maximum graphs with no vertex of degree $t$ into each of the partition classes, and you can even prove that all extremal graphs have this structure.