Sent emails not showing on iMac but do on iPhone

Thanks, I suspect Your Mac is short on resources, but we can check on that better...

Have you run Disk Utility's Disk First Aid on all connected drives?

In Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

Are you running any VPN, Anti-Virus, or Cleaning apps?

We need to see what all is running, a report from this will not display any personal info...

Using EtreCheck - Apple Community

EtreCheck is a FREE simple little diagnostic tool to display the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to help people help you with your Mac. It will not display any personal info.

https://www.etrecheck.com/

Thanks for Old Toad’s etrecheck instructions…

Slow iMac 2017 - Apple Community

Just checking, are they both IMAP accounts?

Quit Mail on the Mac & login to your WebMail in a Browser, are Sent Mails showing there?