Despite the Conquest many Reachlords feud among each other using the distraction of the Dornish War as well as Tyrell's focus on Aegon's court in King's Landing to settle old scores and reclaim bits and pieces of the many petty Kingdoms that existed before the Gardener Kings.
House Tyrell, stretched too thin and too new to being entirely in control fail to manage these conflicts which lead the Faith Militant to take advantage of the regions uncertainty, building as strong a base in the Reach by fighting against the nobility, as they did in the West by fighting for them.