  The projective norm graphs $\text{NG}(q,t)$ provide tight constructions for
the Tur\'an number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$. In
this paper we determine their automorphism group and explore their small
subgraphs. To this end we give quite precise estimates on the number of
solutions of certain equation systems involving norms over finite fields. The
determination of the largest integer $s_t$, such that the projective norm graph
$\text{NG}(q,t)$ contains $K_{t,s_t}$ for all large enough prime powers $q$ is
an important open question with far-reaching general consequences. The best
known bounds, $t-1\leq s_t \leq (t-1)!$, are far apart for $t\geq 4$. Here we
prove that $\text{NG}(q,4)$ does contain (many) $K_{4,6}$ for any prime power
$q$ not divisble by $2$ or $3$. This greatly extends recent work of Grosu,
using a completely different approach. Along the way we also 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
give a new, more elementary proof for the $K_{4,7}$-freeness of
$\text{NG}(q,4)$.
