When a doctor writes out a prescription for medicine they usually write a lot of prescriptions and notes throughout the day so that is one reason they write horrible as to where it's hard to read the doctors handwriting.
That is what a doctor told me when I asked them about the handwriting and why it was so horrible.
Also some doctors are older and have trouble with their hands as well so that makes it hard to write properly.
Here's an example from a doctor on reddit on why there handwriting is horrible.
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Resident doctor here. My handwriting is pretty legible, as is the handwriting of my other colleagues.
Most of the paperwork and scripts we do are done on the computer now, so we don't have time to practice shitty writing. I notice that a lot of the handwriting I can't decipher is done by the older physicians who had to do a lot more handwritten paperwork back in the day. Since there's a lot of work to do and patients to see, it's faster and more efficient to write abbreviations of mostly everything. That can definitely get confusing since there are multiple ways to abbreviate the same thing.
It's no excuse, but when there's several patients you have to see per hour who all demand more time out of you or try to counter your medical advice with whatever they saw on Dr. Oz, legible handwriting is the least of our worries.
However, when I'm on a time crunch and don't have a computer readily available, I often scribble some notes down quickly before returning to it later. When I do, sometimes I can't even decipher what I wrote down. Then I have to look at it in context of all my other notes on that patient to figure it out.
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That above is not from me but from a post on the website reddit on a question about why the doctors handwriting is not easy to read and why they have bad handwriting.
Now a lot of doctors are doing prescriptions through the computer so they do not need to write out prescriptions but instead type the prescription out on the computer.