Connect the Toshiba HD. Verify in Finder Settings : General panel that hard disks and external disks is selected. Boot your 24 inch iMac into Recovery and run Disk Utility. On the latter's left panel, select your Toshiba HD if shown, and perform First Aid on it. If you do not receive a successful green result, perform First Aid again on it. If you can not get a successful First Aid pass on the drive, or it never appears in the side panel, then quit Disk Utility and reboot the iMac. You are now in the market for a new Time Machine drive.
I happen to use Crucial X8 SSD drives exclusively on my Macs for Time Machine backups. The drive should have capacity that is 2.5 - 3x your Internal iMac drive size. You would mount the new drive to the Desktop, launch Disk Utility, select the drive name in Disk Utility's side panel, and then Erase. I tend to name my new Time Machine disks as TM8_operating system (e.g. TM8_Ventura). You want to format the entire drive as APFS.
You then need to open System Settings : General : Time Machine > and add + the new drive. Under Options… you want to set the backup frequency to Automatically Every Hour. Your first backup to the new drive will be a full backup and with that Crucial drive, will take only a few minutes instead of hours to that much slower Toshiba HD.