Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$.
Assume $g=g(X) \geq 2$.
Is the degree of $X\to Y$ bounded by $84(g-1)$?
I think the answer is yes. In fact, the degree of $X\to Y$ is the cardinality of $\# \mathrm{Aut}(X/Y)$. This is bounded from above by $\# \mathrm{Aut}(X)$ and this is known to be bounded from above by $84(g-1)$ (if $g>1$).