  An $(n,s,q)$-graph is an $n$-vertex multigraph in which every $s$-set of
vertices spans at most $q$ edges. Tur\'an-type questions on the maximum of the
sum of the edge multiplicities in such multigraphs have been studied since the
1990s. More recently, Mubayi and Terry [An extremal problem with a
transcendental solution, Combinatorics Probability and Computing 2019] posed
the problem of determining the maximum of the product of the edge
multiplicities in $(n,s,q)$-graphs. We give a general lower bound construction
for this problem for many pairs $(s,q)$, which we conjecture is asymptotically
best possible. We prove various general cases of our conjecture, and in
particular we settle a conjecture of Mubayi and Terry on the $(s,q)=(4,6a+3)$
case of the problem (for $a\geq2$); this in turn answers a question of Alon. We
also determine the asymptotic behaviour of the problem for `sparse' multigraphs
(i.e. when $q\leq 2\binom{s}{2}$). Finally we introduce some tools that are
likely to be useful for attacking the problem in general.
