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Erd\\H{o}s and Simonovits \\cite{ESi1} conjectured that for every family $\\mathcal{F}$ of bipartite graphs, there exists $k$ such that $\\ex{n}{\\mathcal{F} \\cup \\mathcal{C}_k} \\sim \\ex{n}{\\mathcal{F} \\cup \\mathcal{C}}$ as $n \\rightarrow \\infty$. This conjecture was proved by Erd\\H{o}s and Simonovits when $\\mathcal{F} = \\{C_4\\}$, and for certain families of even cycles in \\cite{KSV}. 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