  Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom
graphs contain asymptotically the minimum number of copies of $H$ taken over
all graphs with the same order and edge density. While still open for graphs,
the analogous statement is known to be false for hypergraphs. We show that
there is some advantage in this, in that if Sidorenko's conjecture does not
hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is
possible to improve the standard lower bound, coming from the probabilistic
deletion method, for its extremal number $\mathrm{ex}(n,H)$, the maximum number
of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this
application in mind, we find a range of new counterexamples to the conjecture
for hypergraphs, including all linear hypergraphs containing a loose triangle
and all $3$-partite $3$-uniform tight cycles.
