You and your mount act on different (but co-current) turns.
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
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.
Cavalry Training and Trampling charge
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.