  Given a family $\mathcal{F}$ of bipartite graphs, the {\it Zarankiewicz
number} $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$
bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a
subgraph (such $G$ is called {\it $\mathcal{F}$-free}). For $1\leq
\beta<\alpha<2$, a family $\mathcal{F}$ of bipartite graphs is
$(\alpha,\beta)$-{\it smooth} if for some $\rho>0$ and every $m\leq n$,
$z(m,n,\mathcal{F})=\rho m n^{\alpha-1}+O(n^\beta)$. Motivated by their work on
a conjecture of Erd\H{o}s and Simonovits on compactness and a classic result of
Andr\'asfai, Erd\H{o}s and S\'os, in \cite{AKSV} Allen, Keevash, Sudakov and
Verstra\"ete proved that for any $(\alpha,\beta)$-smooth family $\mathcal{F}$,
there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large
$n$, any $n$-vertex $\mathcal{F}\cup\{C_k\}$-free graph with minimum degree at
least $\rho(\frac{2n}{5}+o(n))^{\alpha-1}$ is bipartite.
  In this paper, we strengthen their result by showing that for every real
$\delta>0$, there exists $k_0$ such that for all odd $k\geq k_0$ and
sufficiently large $n$, any $n$-vertex $\mathcal{F}\cup\{C_k\}$-free graph with
minimum degree at least $\delta n^{\alpha-1}$ is bipartite. Furthermore, our
result holds under a more relaxed notion of smoothness, which include the
families $\mathcal{F}$ consisting of the single graph $K_{s,t}$ when $t\gg s$.
We also prove an analogous result for $C_{2\ell}$-free graphs for every
$\ell\geq 2$, which complements a result of Keevash, Sudakov and Verstra\"ete
in \cite{KSV}.
