Good question!
It depends on the overlord in question, how strong or weak they are – hence Tytos’ weakness preventing him from squashing feuds among his subjects – but also their governing philosophy. For example, Harwyn Hardhand actively encouraged feuding to keep his subjects too busy fighting eachother to fight him, whereas Harren the Black came down on them with fire and sword.
As for cases of royal intervention, I think it comes down to when the violence is seen as counterproductive enough to warrant the cost of intervention. So for example, the reforming King Aegon V saw the violence in the west as compromising his larger mission of improving the lot of the smallfolk so went into the Westerlands repeatedly, whereas Aerys I didn’t seem to care at all that Dagon Greyjoy was raiding the North, the Westerlands, and the Reach.