Balanced supersaturation for some degenerate hypergraphs

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 $C_{2k}$. Recently, Morris and Saxton established a balanced version Simonovits' theorem, showing such has $C_{2k}$, which are `uniformly distributed' over edges $G$. Moreover, they used this result to obtain sharp bound on number $C_{2k}$-free graphs via container method. In paper, we generalise Morris-Saxton's results for even cycles $\Theta$-graphs. We also prove analogous complete $r$-partite $r$-graphs.