That seems like an odd and unreasonable process to me.

It's unreasonable to expect you to take legible verbatim notes on spec, and it's unreasonable to ask an employee to agree as accurate notes that may not be entirely legible or comprehensible.

If your employer persists in requiring this as part of the process, then your best option is to speak up regularly.

"Can we pause a moment, please for me to catch up. I'd like to read back some of my notes to you to check that you are happy that these are an accurate account."

If they believe that a verbatim record of the meeting is so essential, they should make a digital audio recording. But this is ludicrous overkill for all but the most disputative of circumstances. My approach has always been to simply write up a concise summary of the key points afterwards for the employee to sign.

In reply to Robey:

I would add that if typing works better then you say that you will go and print it and ask that the individual signs the first draft which you then tidy up and be prepared to explain any tidying/clarification

As Robey says their process is just weird anyway but find a workaround

I think this is another version of fast, cheap, good triangle.
The business needs to pick which 2 are most important out of neat, being signed on the day and being accurate (or even verbatim).

In terms of simple tips for you, I've just got a few:
Splitting the page down the middle, one side for manager one for employee, makes it clearer who is speaking and for me is quicker than writing intials.
A good pen that makes writing easier - Uniball eye Micros are the only thing that makes my handwriting legible.
As Robey says feeling confident to pause meeting to catch up is important - these notes are for everyone in the room and they should respect that.
If it's a planned meeting ask the chair/manager/lead for their questions or script (if that prescriptive) in advance, then it's part written for you.
Confirm if it needs to be summary points or verbatim - if you have the choice summarise.

But really you should not need to take notes in this manner.
Hopefully your employer will change the process to be more reasonable in the future.

In reply to Briony C:

Thank you all for your replies. Made me feel a lot better about finding this process odd. I have spoken with my line manager today (we both started on the same day, so a new process to us both) she agreed it’s fine for me to use a laptop and was happy for it not to be on the form, just attached to it and signed off.
The forms are standard and used for anything from a recorded conversation to a disciplinary (with tick box options).
I will definitely make sure to stop them to catch up and have spoken with the shift manager conducting and asked him to watch his pace.
Will definitely take everybody’s advice on board.
Thank you again.