I have seen quite a few cases of very poor handwriting among children who were moved to cursive very early.
My other bugbear is that kids use it to mask spelling issues--if you don't know a word, just make your writing a bit scribbly and the teacher probably won't notice. This means you (the parent or teacher) don't get feedback about which patterns the child need to practice. It also becomes an engrained habit in the child after a while.
Also, what a PP said about the disconnect between the printed and written word. As children enter the free-reader stage, they start doing that thing of "I'm not sure how to spell this word, so I'll try it out on my notebook a couple of ways and see which way looks 'right'" Smart tactic. Except that it does not work very well if they are writing in cursive, as the word looks so different!
It's worthwhile developing a neat, working print style as well as a good cursive style. Apart from anything else, people in some countries do not learn cursive, so you will need to print if you want to write anything handwritten that people from various countries will definitely be able to read.
If you need to move back to print or semi-cursive for a while, you might need to have a discussion with the teacher and be a little bit assertive (but polite, of course) about it. "We think we need to move back to print and develop a neat print style. We will move back to cursive when spelling and handwriting have improved." My daughter went to a school that taught cursive from age 5. It was too soon for her. Instead, we did those "semi-cursive" letters with lead-ins and lead-outs ("flicks") and I said very explicitly to the teacher, "I am sorry, but I am afraid I am going to do it this way with my daughter for the moment and will not be getting her to switch to full cursive until she is ready--trying to make her join right now is causing her stress, which isn't OK."
OP, one thing that might help is copying. Like, literally get a book out, read it through together, then copy it in neat handwriting. If he does not need to think about spelling or content at all, he can focus fully on just the handwriting bit. He can then start slowly working on applying his skills to his independent writing.