I found this tip mentioned in a YouTube comment (though I can't find it now) after trying a handful of other methods (including a clothing iron, etc). All you need is a couple pieces of drywall and sheets of newsprint (probably optional).
EDIT: Newsprint is probably a bad idea (see comments). Any paper that is acid/ink free will probably do the job. I've edited the instructions below to say 'paper' instead of 'newsprint'.
This is the process that's worked for me:
1. When a print is complete and ready to be dried, hang it for just a minute or two to allow excess water to drip off.
2. Place one piece of drywall on a flat surface, white side up (the side without any kind of backing). The sheet of drywall should be large enough that your print will fit on it with at least a couple of inches around the border.
3. Place a sheet (or sheets) of paper over the drywall. Make sure it is very flat without any creases or wrinkles.
4. Place your print on the paper and then either fold the paper over it or place another sheet of paper on top of it (depending on the size of your print).
5. Place a second sheet of drywall, white side down (facing the print), on top of the newsprint.
6. Leave everything as it is for at least a day. After a day, check the print. If it is no longer actively damp, I remove the paper and put the print back between the drywall sheets.
7. Leave the print between the sheets of drywall until it is
completely dry (if it is cool to the touch, it's not dry). This typically takes 3-5 days for me (including the day with the paper), but your experience may vary depending on temperature and humidity.
If you remove the print and it starts to bend/curl over time, it probably wasn't fully dry when it was removed. When done properly, my prints have come out as flat as a new piece of paper. Yes, it's not the fastest method... but I feel like that's a fair tradeoff for being easy, cheap, and working consistently.
These are the drywall sheets I use:
https://www.homedepot.com/p/USG-She...-in-Patch-and-Repair-Drywall-141133/202353426 They're already cut to a size that's easy to work with. That said, you could probably find old scraps of drywall for free... they just need to be clean.