Battle on the Big Bridge was added in patch 2.2 of the A Realm Reborn expansion.
Battle on the Big Bridge becomes available at level 50 as part of the “The Three Collectors” quest, which is given by Hildibrand – Western Thanalan (X:12, Y:13).
The trial has a minimum ilevel of 50 to enter and rewards you with:
Allagan Tomestone of Poetics: 10