Very broad question, I know, but the Dope has oft pleasantly surprised me so I’m hoping for something good.
What I would like to know is what evidence there is about whether, how, and how easily an adult who can’t write non-fiction very well can learn to write non-fiction fairly well.
In other words, an adult who can not organize an essay, express thoughts accurately, write logically, write grammatically, etc learns, over time, to do all these things.
Even more impressionistically: An adult goes from writing prose that seems profoundly unintelligent to learning to write prose that sounds at least competent.
When this happens (if it happens!) how does it happen?
Evidence: Joseph Conrad first published after 40 years of age. He was a fiction writer, but it is harder to do than non-fiction.
There is an irritating myth that writing is a special gift or skill, which some people are born with; ‘the myth of facility’. Like any other skill it takes time and practice.
But what do we know of how well he wrote at 20, and how he got from there to where he was at 40?
Right–the motivation for my post is a little sad. I believe it’s a skill that takes time and practice. But experiences are bumming me out on this and making me suspect some people just “don’t got it.” Not necessarily that they were doomed from birth, but that if you don’t learn it by a certain stage of mental development, you’re just not going to learn it.
I’m hoping to cheer myself up with studies and reliable anecdotes to the contrary. 
Time, practice, feedback, and large amounts of exposure to writing of the standard you wish to attain. The last part, especially.
Like anything, some people have more affinity for the task than others. But any person can attain a workmanlike level with the application of time, practice, feedback, and reading. I know plenty of people from law school for whom English was a language they learned later in life, who achieved a perfectly acceptable standard.
OP, if you’e not getting one-on-one feedback, if you’re just practicing alone, you’re obviously going to be frustrated with progress. Like anything with an element of subjectivity, you can’t progress without knowing where and how you are going wrong. Fundamentally, non-fiction does not have to be fancy. Mechanical approaches (explicit instructions for what to put in each paragraph) can be very helpful until you internalize organization.
I can’t offer any studies or research on this, but will offer a few thoughts.
I think writing is a matter of imitation. Kids are born imitators, which is probably why they learn languages and writing so much more easily. Even as an adult, though… when I read, I am constantly analyzing the author’s choice of style, word, structure, etc. For example, reading Stephen King showed me that a great way to make the setting seem more believable and real is to insert some random bit of information that is totally irrelevant to anything, but would nonetheless be something you’d know/notice if you were there. (In one short story, a street is named and it is mentioned how the name is pronounced differently from the spelling).
To take imitation a step further, you can take a passage of text and write about something entirely different, but using the same structure - almost like a thorough ad-libbing. Eventually, you’ll realize that there’s another myth about writing - the myth that says each writer has their own voice, as if there is just one per writer. That’s silly… writers have a different voice for each purpose, just like you talk different in a library than in a bar.
The real problem is that very few people truly value writing skills. For most adults, the only time they consciously practice writing is when putting a cover letter and resume together.
to make this question meaningful you would need to control for the initial IQ. If the guy has high inborn intelligence but has not learned to write well (maybe never learned to write at all, maybe he is an illiterate prisoner in Martian captivity since childhood or something like that) one would expect that once you teach him to write you can also teach him to write well.
By contrast, if he is generally known to have low intelligence and got nowhere with writing despite much educational opportunity in the past, then a sudden breakthrough improvement in quality would be much more surprising.
I can’t offhand think of any Dopers who have gotten noticeably better at expressing themselves over the years they’ve been here, but if there were, that would be good anecdotal evidence.
I think dracoi’s point about imitation is a good one, and that people who can’t express themselves well in writing are people who haven’t done much reading. If they consciously made an effort to read more, would they be able to write? I don’t know.
I suspect the best person to address the OP’s question would be someone who teaches developmental/remedial writing at a community college.