From "Controlling a Mount" on p.198 of the Player's Handbook:
The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options; Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.
Does this mean that the rider and mount share a single turn?
The key is in the passage you quote:
It moves as you direct it, ...
"... as you direct it ..." includes both spacial and temporal directions - "move here now, move there later, move back here after that" only, you know, using reins and things instead of words.
You direct the movement of the mount and can take your action and possible bonus action at any time during that movement. The mount can also take its action (limited to Dash, Disengage and Dodge) and possible bonus action at any time during that movement. Note that the limitation on actions does not affect bonus actions.
If you are lucky enough to have a mount with lair or legendary actions (go the dragon rider!) they happen when they otherwise would.
Initiative determines the order of turns during combat. [...] The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round.
The PHB describes the rules for mounted combat. In them it describes that a controlled mount has an initiative that is the same as the rider's:
The initiative of a controlled mount changes to match yours when you mount it.
So, what does "changes to match your [initiative]" mean? It means that they have the same total dex check score that you got (after rolling a d20 and adding bonuses). That means that you are tied for initiative. Luckily, there are rules specifically for the case of initiative ties:
If a tie occurs, the DM decides the order among tied DM-controlled creatures, and the players decide the order among their tied characters. The DM can decide the order if the tie is between a monster and a player character.
So, if two creatures have the same initiative (a tie) the players/DM get to decide in which order they go. Nowhere does it say that creatures with the same initiative go on the same turn. And nowhere in the rules for mounted combat is this suggested either.
Thus, the mount gets its own turn but the rules make it so that it acts on a turn adjacent to yours.
All this ruling means is that mount and rider take separate turns. The mount can move and act then the rider can move and act (or vice-versa). This is the exact way that independent mounts clearly work and all this answer is saying is that it is also true with controlled mounts.
Jeremy Crawford has clarified and agreed with this here:
Q: I listen your podcast Dragon Talk: seems like you said that you and the controlled mount have the same turn. Does this mean that the rider and mount share a single turn? [...]
A: A controlled mount has its own turn, but that turn takes place on the same initiative count as the rider’s turn.
A rider and a controlled mount have separate turns, but they have the same initiative, which means you decide which one goes first.
It is fairly common, in my experience, to house rule that the rider and mount can interweave their movements and actions. I don't see, and have not experienced, any issue with this as far as game balance or combat flow is concerned. So, if a DM wished to play it like this there shouldn't be a huge issue. However, from the rules, it is clear that this is not the intended interpretation.
Unlike in previous editions of D&D, a creature's initiative in 5e is an ordinal (e.g "third turn in the round") rather than a numerical score (e.g. "Dexterity check total of eleven").
Because initiative is an ordinal, the "Controlling a Mount" rule works as follows:
A combat is in progress. Applejack the Warhorse's initiative is the sixth turn of every round. Bob the Battlemaster's initiative is the third turn of every round.
If Bob were to be unhorsed before the end of the combat, Applejack's initiative would revert to sixth again; the initiative order itself never changes.
Since everything hinges on the phrase "the initiative of a controlled mount changes", let's comb through the rules text.
From page 189 of the Player's Handbook:
Initiative
Initiative determines the order of turns during combat.
Tells us what initiative does, but not what it is.
When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. [...] The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.
There are two distinct concepts here:
The Dexterity check total. This is a numerical value in (roughly) the 1-20 range. There can be ties.
The rank in the initiative order. This is an ordinal value (first, second, third,...last). There are never ties, since they're always resolved according to the following rule:
If a tie occurs, the DM decides the order among tied DM-controlled creatures, and the players decide the order among their tied characters. The DM can decide the order if the tie is between a monster and a player character.
Taking (2) above as the definition of "initiative" gives the most satisfactory interpretation: initiative is a creature's rank in the initiative order. Creatures only share initiative in two cases:
It means that they share an initiative, you pick who goes first. Technically at the beginning of when this happens, and then it stays that way until the combat is over or the ridership is transferred, since there is no delaying by RAW.
I'd imagine however, that many games house rule this so that they can take their turns interspersed.
PHB pg 198
CONTROLLING A MOUNT
While you're mounted, you have two options. You can either control the mount or allow it to act independently. Intelligent creatures, such as dragons, act independently.
You can control a mount only if it has been trained to accept a rider. Domesticated horses, donkeys, and similar creatures are assumed to have such training. The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options; Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.
An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes. It might flee from combat, rush to attack and devour a badly injured foe, or otherwise act against your wishes.
In either case, if the mount provokes an opportunity attack while you're on it, the attacker can target you or the mount.
PHB pg 311
TRAMPLING CHARGE (warhorse)
If the horse moves at least 20 feet straight toward a creature right before hitting it with a hooves attack, the target must succeed on a DC 14 Strength saving Throw or be knocked prone. If the target is prone, the horse can take a bonus action to make another attack with its hooves against the target.
OotA pg 226
CAVALRY TRAINING
When the Chitik hits a target with a melee attack while mounted on a creature, the mount can make a melee attack against the same target as a reaction.
There are two cases:
When you mount an unintelligent creature who qualifies as a mount, it's previously independent initiative (as a separate creature) changes to match yours, if you choose to control it. If you choose to allow it to act independently, then it's initiative does not change. (note the wording above: "The initiative of a controlled mount changes")
In the case where you take control, the two of you move co-currently. It does not lose it's initiative, rather it simply matches yours, and acts co-currently to your turn. If you were to dismount, it would still act on it's new initiative, which is equal to the former riders (if not, it could act twice in one round, by rolling initiative lower than it's riders, and then having the rider dismount).
When you mount an intelligent creature who qualifies as a mount, it retains its place in the initiative order. You do not direct it's actions, and your presence does not limit it's ability to act.
From the above analysis, the two creatures do not share a turn. Thus if a Chitik riding a warhorse were to move 20 ft straight toward a creature, and then attacked them, the mount could make a single hooves attack, but not use the "trampling charge" feature.
Furthermore, the mount would have to move away, back again, and then make a second hooves attack before using the trampling charge feature.
This may be best understood in the following (legal) example:
Cor the Chitik Cavalier is riding Bree the warhorse. Cor&Bree start 15 ft away from a goblin Droop.
Cor directs Bree to move 5 ft towards droop to be within range.
Cor attacks Droop with his Spear. Because of his Cavalry Training feature, Bree can move 5ft closer and make an attack on Droop with a reaction.
Cor then directs Bree to move 20 ft away from droop (provoking an opportunity attack).
Cor and Bree move 20 ft towards Droop.
Bree makes a hooves attack as an action. Droop fails his STR ST, and falls prone.
Bree makes a bonus action hooves attack on Droop.
Cor and Bree use the last of their movement and move 10 ft away from Droop.
