  The odd-ballooning of a graph $G$, denoted by $G_q$, is the graph obtained
from replacing each edge in $G$ by a odd cycle of the same size where the new
vertices of the odd cycles are all different. In 2002, Erd\"os et al.
determined the extremal graphs of $k$-fan. In 2016, Hou et al. determined
extremal graphs of the odd-ballooning of stars for $q\geqslant 5$. In 2020, Zhu
et al. determined extremal graphs of the odd-ballooning of paths for
$q\geqslant 3$. In this article, we use progressive induction lemma of
Simonovits to determine the extremal graphs of both odd-ballooning of stars and
odd-ballooning of paths for $q\geqslant 3$.

Combinatorics