Jeremy Crawford continues to confuse me. :)
A rider and a controlled mount have separate turns, but they have the same initiative, which means you decide which one goes first.
But in Dragon Talk interview with him in February 2018 about mounted combat (the relevant portion starts at 19:10), he says:
If you control the mount, the creature's initiative changes to your initiative. You're now acting as a unit. It still has a turn, but its turn basically overlaps with yours. It gets its move. And [...] it's moving on your turn, so then far easier for your character to coordinate with the mount. Its movement is taking place on your turn, and its action options are limited [...] its only actions are Dash, Disengage and Dodge. And so that means it's not attacking [...] The fact that it can Disengage means it can move without triggering opportunity attacks, and the beauty of it acting on your turn, [...] its turn overlapping with yours, is that then also your movement is still free to use on your turn and all your actions are still available.
So the mount almost becomes a movement and action extension for the rider. So that's a really powerful advantage. Even though the mount is giving up things like attacking and whatnot, you're gaining on your turn all this potential extra movement, and also basically a free - for the mount, at least - Disengage, Dodge, or Dash, and Dash means even more movement.
This contradicts his earlier tweet, and goes further than the PHB.