Let $X \to \mathbb P^n$ be a fiber bundle of algebraic varieties with $X$ an affine variety. What is the smallest dimension that $X$ can be?

An obvious lower bound is $n+1$.

An upper bound is $2n$, given by taking the complement of a generic effective divisor of bidegree $(1,1)$ in $\mathbb P^n \times \mathbb P^n$. (Alternatively, take the canonical "duality" $(1,1)$ divisor defined by viewing $\mathbb P^n$ as the Hilbert scheme of hyperplanes on $\mathbb P^n$.) The complement of an effective ample divisor is of course affine, and since the divisor is locally constant on the fibers its complement is as well.

This construction is the construction in this question, which also inspired my question, with the second condition removed.