Fact 1 (The Hurwitz Bound): If $X$ is a smooth projective connected curve of genus $g\ge 2$ over $\mathbf{C}$ then
$$| Aut_{\mathbf C }(X)| \le 84(g-1)$$

Fact 2: $Aut_\mathbf{C}(X) = Aut_{\overline K}(X)$ (The sentence that says "If $\phi$ is an automorphism of $X$ then $\phi$ must be one of these possibilites" is first order and thus by the first order completeness of algebraically closed fields of characteristic zero...)

Fact 3 (See Silverman's Arithmetic of Elliptic Curves chapter 10 or Serre's Galois Cohomology or Berhuy's notes or...): The twists $W_{/K}$ of a variety $V_{/K}$ are given up to isomorphism by the pointed set $H^1(Gal(\overline K /K),Aut_{\overline K}(V))$. Assume now that $L/K$ is Galois. If not you can just replace $L$ by its Galois closure. The twists which resolve over $L$ are given up to isomorphism by $H^1(Gal(L/K),Aut_{\overline K}(V))$

Fact 4 (Exercise): $$ |H^1(Gal(L/K),Aut_{\overline K}(V))| \le 84(g-1) | Gal(L/K)| $$

I must say however that I'm not sure what sections have to do with anything.