This includes several of the in-instance dungeon quests like “Saving the Botanica” that pop into your quest log by “radio” and you turn them in using the quest log. There is no way to progress to e.g. “Culling the Herd” and “A Most Somber Task”. It seems the turn-in behavior was not copied/implemented in Cata Classic. Since this mechanic is used often through the Cata leveling zones, fixing it is probably important.
I’ve had several of these work fine, make sure you don’t have addons interfering with it, and if that’s not it, then it’s probably only individual quests, so make note of which ones you encountered issues with.