The Zilber-Pink conjecture is a far reaching and widely open conjecture in the area of unlikely intersections generalizing many previous results in the area such as the André-Oort conjecture. Through variations of a strategy first introduced by Pila and Zannier, one can reduce this conjecture for curves in certain Shimura varieties to upper bounds for the size of Galois orbits of so called "atypical points" on such curves. We discuss how recent advances in a method first introduced by Y. André that produces height bounds help us establish these upper bounds for Galois orbits, and thus cases of Zilber-Pink, for certain curves in spaces such as $Y(1)^n$ and $\mathcal{M}_g$.