  A classical theorem of Simonovits from the 1980s asserts that every graph $G$
satisfying ${e(G) \gg v(G)^{1+1/k}}$ must contain $\gtrsim
\left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$. Recently, Morris and
Saxton established a balanced version of Simonovits' theorem, showing that such
$G$ has $\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k}$ copies of $C_{2k}$, which
are `uniformly distributed' over the edges of $G$. Moreover, they used this
result to obtain a sharp bound on the number of $C_{2k}$-free graphs via the
container method. In this paper, we generalise Morris-Saxton's results for even
cycles to $\Theta$-graphs. We also prove analogous results for complete
$r$-partite $r$-graphs.
