does your vocal range improve when you end puberty?

With professional training, yes, your range could sing falsetto. But normal people usually lose range after puberty.

While his voice is still "cracking", he doesn't even HAVE the ability to control a lot of his pitch. He is comparing himself to an ADULT singer. It is possible to take voice lessons through his awkward stage, but he will NOT be trained to sing like he unrealistically expects to--it would destroy, not improve his voice. Learning to sing in falsetto is possible only when his voice finally matures, and even then falsetto singing may not suit him at all.

The worst thing a YOUNG amateur singer can do is to compare his voice to an ADULT who has sung longer than the young singer has even LIVED. Thom Yorke is now 48 years old, and possess a very unusually wide vocal range, so not many adults can sing like him. It's best to just grow up and then find out what your own voice will and won't do. Why ruin a potentially gorgeous baritone or bass voice because you falsely believe that singing high notes make you "talented".

The only proper solution to this is to start taking OFFLINE one-on-one lessons with a GOOD vocal teacher.