  A classic result of Erd\H{o}s and, independently, of Bondy and Simonovits
says that the maximum number of edges in an $n$-vertex graph not containing
$C_{2k}$, the cycle of length $2k$, is $O( n^{1+1/k})$. Simonovits established
a corresponding supersaturation result for $C_{2k}$'s, showing that there exist
positive constants $C,c$ depending only on $k$ such that every $n$-vertex graph
$G$ with $e(G)\geq Cn^{1+1/k}$ contains at least
$c\left(\frac{e(G)}{v(G)}\right)^{2k}$ many copies of $C_{2k}$, this number of
copies tightly achieved by the random graph (up to a multiplicative constant).
  In this paper, we extend Simonovits' result to a supersaturation result of
$r$-uniform linear cycles of even length in $r$-uniform linear hypergraphs. Our
proof is self-contained and includes the $r=2$ case. As an auxiliary tool, we
develop a reduction lemma from general host graphs to almost-regular host
graphs that can be used for other supersaturation problems, and may therefore
be of independent interest.
