You'd think the dragons would have deterred them, but that isn't why, though Aegon's son Maegor the Cruel did attack them with Balerion more than once -- something about the succession, and about him having more than one wife. They all fought ordinary battles too, with men on the ground.

Maegor's successor Jaehaerys made peace with the Faith Militant by pardoning the ones who would lay down their swords. The Faith only had to acknowledge the Targaryen marriages, they didn't have to approve of them.

[He shakes his head, blinks, and half-chuckles.]

My old Maester would be pleased that I can remember all that. I had other heroes as a boy, not Maegor or Jaehaerys.