We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of in-equivalent height functions on ℙ1(ℚ̄). 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 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 ℂ, and a uniform Bogomolov bound for the family over ℚ̄.