So in other words, you stop the barbarian from completing his entire move because he doesn't have enough movement to cross the chasm.
I don’t know what you mean by “completing his entire move”. If you mean that he isn’t moving a full 60 feet, I don’t understand. There are always situations in combat where a player decides not to use every last foot of his or her character’s movement. For example:
DM: The orcs look like they’re spoiling for a fight. What do you do?
Player: I close the distance between us and attack.
DM: Okay, they’re about 20 feet away. <Resolves attacks> Two orcs fall dead. You have 10 feet of movement left if you want to use it, but moving away will incur opportunity attacks from the remaining orcs.
Player: Hey! Why are you stopping me from completing my entire move?
DM: ???
I bend the rules now and then to fit the narrative of the moment. Someone wants to swing from a chandelier, dive sword-first onto a monster or leap and grab on to that dragon? I'll make it work even if the letter of the rules says it doesn't work. Not everything the player attempts will work of course, it has to be logically consistent with D&D's action-movie universe. That's just fundamental to how I run my games. I think the guy leaping through the air while we cut to the bad guys is more dynamic and fun.
Okay, but that’s just putting a narrative you prefer over a narrative you don’t prefer.
I'm not saying putting rules first, narrative second is wrong because if you and yours are having fun it's perfectly okay. It just bugs me that you won't just admit that you stop the barbarian from using the remainder of jumping because of your ruling. Or conflating your ruling somehow with character effectiveness, which to me is like saying the sky is blue because the ocean is wet.
I don’t see a conflict between the rules and the story the way you seem to. To me, it isn’t a zero-sum game. The rules are an input into the resulting narrative. I don’t stop the barbarian from using his remaining 10 or 15 feet of movement. He could make a detour on his way to the chasm or move to a different point along its edge. He could even use his remaining movement to hurl himself into the chasm if he wants to. But if his goal is to leap to the other side, and he doesn’t have enough movement to get there, then that declaration and the resolution of that action are going to have to wait until he does have enough movement.
You've decided that movement ends at the end of a turn and you can't end a turn mid-air. That's all you have to say.
It’s simpler than that.
First, I should think that your turn is when you get to decide how to use your movement, so of course when your turn ends your move has ended as well. But if you’re saying that I don’t consider movement
in the fiction continuous from turn to turn, then no, I absolutely do. From a character’s perspective, the end of its current turn and the beginning of its next turn are pretty much a single instant in time. The barbarian could certainly move the obligatory 10 feet at the end of his turn and make the jump at the beginning of his next turn in one of my games because that’s what he was doing immediately before jumping.
As for my ruling (which I believe is consistent with official rulings on the matter of jumping), it’s that if your declared action for your turn requires movement, you must have enough movement on your turn to complete the declared action. The barbarian doesn’t have enough movement to complete the jump, so the jump can’t be part of his declared action for that turn.