The projective norm graphs $\NG(q,t)$ provide tight constructions for the Tur\'an number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$. The determination of the largest integer $s_t$, such that the projective norm graph $\NG(q,t)$ contains $K_{t,s_t}$ for all large enough prime powers $q$ is an important open question with far-reaching general consequences. Here we settle the case $t=4$. Along the way we also develop methods to count the copies of any fixed $3$-degenerate subgraph, and find that projective norm graphs are quasirandom with respect to this parameter. Some of these results also extend the work of Alon and Shikhelman on generalized Tur\'an numbers. Finally we also completely determine the automorphism group of $\NG(q,t)$ for every possible values of the parameters.