It's probably the broken crystal
Here's Kylo Ren's lightsaber:
Notice how "jagged" it is.
Now here is Ezra Bridger with a lightsaber of similar design, but much more ancient. This is from Star Wars Rebels.
It might just be a matter of animation quality, but it certainly looks straighter to me.
In any case, according the Force Awakens Visual Dictionary:
Kylo Ren's unusual lightsaber is an ancient design, dating back
thousands of years to the great scourge of Malachor. The crossguard
blades, or quillons, are tributaries of the primary central blade, all
spawning from a cracked kyber crystal that is the cause of their
ragged, unstable appearance.
The cause of the crossguard is different, and has to do with the hilt design.
An array of focusing crystal activators split the energy stream into
perpendicular blade energy channels, creating the quillons.
As for the reason for the broken crystal, we can only speculate. Perhaps Snoke viewed a broken crystal as suitable for an unstable Dark Side user. Perhaps Snoke required that Kylo make his own lightsaber, and it was all he could find.
One thing is certain, it was not due to lack of avaibility. Kylo trained at the
Jedi academy, and presumably had a different lightsaber then, with a functioning kyber crystal.