It’s not necessarily your fault because it’s the way it works in 5e, but I find the mounting rules are a bit janky: the dragon and the rider get to roll separate initiative, but match their initiative the moment the rider gets on top of the dragon, with some edge cases when the two get separated or the dragon gets mounted mid battle.
I propose a far more elegant fix: the two roll their own initiative and act during their turn, as usual; while mounted, the rider moves using the dragon’s speed, but during the second turn (either the dragon’s or the rider’s, depending on who went first in initiative order), the players must subtract from the dragon’s speed the movement it did during the previous turn (so, for example, if the rider moves 20 feet forward using the dragon’s speed; when the dragon’s turn comes, it can move 20 feet less than its total walking speed, because it has already moved 20 feet this round).
This way, you only need to keep track of the movement; turn order remains unchanged. I know it’s a divisive opinion, because some players may find it easier to move the initiative rather than writing down how many feet the dragon walked in a given round; but it could be worth having it as an optional rule. It’s how I run it in my games, and I find it a lot more elegant than having the mount’s turn swing up and down in initiative count whenever someone gets on top of it.