Stability of normal bundles of Brill-Noether curves

We prove that the normal bundle of a general Brill-Noether curve of genus $g \geq 1$ and degree $d$ in $\mathbb{P}^r$ is semistable if $g=1$ or $g\geq \left \lceil \frac{5r}{2}\right\rceil r(r-1)$, or $d$ is larger than an explicit function of $g$ and $r$. We further prove that the normal bundle is in fact stable if $g\geq 2$ and either $g$ or $d$ satisfy slightly stronger bounds. In particular, for each $r$ and $g\geq 1$ (respectively, $g\geq2$), there are at most finitely many $(d,g)$ for which the normal bundle of the general Brill-Noether curve is not semistable (respectively, stable).