You continue your jump on your next turn...
...so long as you aren't knocked prone by an attack or effect before you start your next turn.
- Things don't need to be on the ground when a round ends
DMG, page 110:
"A strong wind ... makes flying by nonmagical means nearly impossible. A flying creature in a strong wind must land at the end of its turn or fall" This implies a creature flying doesn't normally have to end a "round" on the ground, and can instead move forward at speed as if no rounds exist--because rounds don't exist to the perception of characters in the world.
2) There are two answers that seem to conflict, but don't
Both are by Jeremy Crawford, but they don't actually conflict if you understand that "speed", "distance" and "Movement" are not the same things. Speed is "distance over time", the unit of time is one 6-second turn/round, so your character's speed is their movement every six seconds.
2a) Answer 1, September 2014
"Things like the jump spell don't increase speed. You can jump crazy far, but your speed caps it." "Every foot jumped costs movement, so you can jump farther than your current speed if you take the Dash action."
(Jeremy Crawford, September 2014.) He tends to be infuriatingly coy and indirect, not really answering the question. He addresses "speed", but not "distance". Speed is distance over a specified time, so is more specific than distance. I presume he thought the question was, "Can I use all my Movement then Jump to increase distance traveled in a single turn?" instead of "over multiple turns."
2b) Answer 2, August 2017
"A caster **doesn't perceive turns ending".
(Jeremy Crawford, Aug 2017) He said "caster" because the question involved a spell, but the bigger question was triggering a Readied Action (Spell) to go off "at the end of this round". Non-casters can ready actions, so clearly no character can "perceive turns ending". If they conflict, this answer was posted nearly three years later, and the most recent rulings are the prevailing ones. This one feels more relevant to the question anyway.
3) Hover trait
"A flyer that lacks the hover trait can stay aloft without moving each round." (Jeremy Crawford, July 2015)
A flying creature does not need the ability to stay in one place to stay aloft during the transition from one round to the next. It can remain flying as long as it is in control of itself. This indicates that to the flying creature, it's not moving for six seconds then "waiting" then moving again, it's just moving.
4) "Flying Movement"
PHB, page 191:
"If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic" This reinforced point 3, that a creature flying under its own control doesn't have "hiccups" in their movement across turns.
All of the above are consistent with a jump over several turns
If a character doesn't know its turn has ended, it has no reason to curtail its movement options to something that can only occur in a six-second span. So, they can jump further than their movement speed cap, but can only cover it's movement speed out of the total distance covered in a single turn.
If this were otherwise, in no D&D world could anyone throw a ball, shoot an arrow, toss a javelin, etc., if the distance from start to finish took longer than six seconds at its speed of travel: the projectile would simply stop mid-air and drop to the ground. If such a thing occurred, everyone in the world would "perceive turns ending", since anything without flying or under the control of a spell would stop moving and drop to the ground in a steady six-second pulse.
Think about this another way: "turns" don't exist outside of combat. Turns don't exist to characters at all. They only exist in the minds of players. If anyone or anything in their world could ever make a long jump or long throw (like archers firing arrows at a high angle to increase range), but couldn't inside combat, they would be aware of the six-second "turn/round" pulse, and that doesn't really match anything in the lore, rulebooks, or novelizations, and doesn't seem at all to match the intent of the game.