  We consider simply-connected $4$-manifolds admitting Lefschetz fibrations
over the $2$-sphere. We explicitly construct Lefschetz fibrations of genus $3$
and $4$ on simply-connected $4$-manifolds which are exotic symplectic
$4$-manifolds in the homeomorphism classes of $\mathbb{C}
P^{2}\#7\overline{\mathbb{C} P^{2}}$ and $\mathbb{C}
P^{2}\#8\overline{\mathbb{C} P^{2}}$, respectively. From these, we provide
upper bounds for the minimal number of singular fibers of such fibrations. In
addition, we prove that this number is equal to $14$ for $g=2$. When such
fibrations are hyperelliptic, we prove that it is $18$ for $g=3$. Moreover, we
discuss the number for higher genera.
