"His teacher has just asked him to write more, which I don't think is very helpful, as it hasn't pinpointed any particular areas he's weak on."
We have had exactly the same comment. Asking for "more output" is hardly a measurable target. My son's English is nowhere near as your as your son's. We found that getting to concerntrate on sentence stucture has improved the amount he writes.
Boring sentence
We went shopping.
Slightly more interesting.
We went late night shopping in the new out of town shopping centre
adding a connective and a couple of adjectives
We went late night shopping in the new out of town shopping centre although it was busy we manged to buy everything we needed.
Changing the sentence structure andtrying to include more interesting vocab
Going late night shopping in the new out of town retail park before Christmas was an arduous experience; everywhere was bustling with crowds nevertheless we bought everything we needed.
Incidently English was never my strong point so please don't tear my writing apart. I am sure that your son is doing stuff like the above in school
Writing is very much an art. It is not easy to level writing. Sometimes writing that achieves a level 5/6 may well be very boring and sometimes a level 4 piece of work may well be more fun to read. Getting a high level in writing seems to be very much a tick box exercise rather than communicating ideas.
There is no way you want all your sentences to be long. There is also the art of structuring writing and writing for your audience.
Maybe your son just needs time to be and develop.