Consider the projection map $$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$ where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of some degree $d$ such that $V(f) \subset \mathbf P^n$ is smooth and $g$ and $h$ satisfy $\deg g+\deg h = d$ and they similarly define smooth hypersurfaces in $\mathbf P^n$ such that $V(gh) \subset X$ is a normal crossings divisor, i.e., $V(g) \cap V(h)$ is again smooth of the expected dimension.

As the fiber $V(f) \cong \pi^{-1}([1:0])$ is smooth by assumption, the map is generically smooth, so has only a finite number of singular fibers.

Question: Is there an upper bound on the number of singular fibers of $\pi$? (In terms of the degree of $f$, for example?)

(If there is an answer to a similar question for more general families of hypersurfaces, I am interested in this, too.)

Thank you!