  We introduce a general strategy for proving quantitative and uniform bounds
on the number of common points of height zero for a pair of inequivalent height
functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to
prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on
the number of common torsion points of elliptic curves in the case of two
Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform
bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford
bound for the family over $\mathbb{C}$, and a uniform Bogomolov bound for the
family over $\overline{\mathbb{Q}}.$
