Given a family
$\mathcal{F}$
of bipartite graphs, the 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
$\mathcal{F}$
-free). For
$1\leq \beta \lt \alpha \lt 2$
, a family
$\mathcal{F}$
of bipartite graphs is
$(\alpha,\beta )$
-smooth if for some
$\rho \gt 0$
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 Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte 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
$\delta \gt 0$
, 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 Verstraëte.