How to update an older iMac?

+1 to running EtreCheck for a comprehensive diagnostic.

However if the problem is "super slow" then I would bet you have a 1TB 5400rpm HDD. Those spinners are too slow to run Catalina. The solution is to upgrade your boot drive to an SSD, which can easily be done by purchasing a USB 3.0 external SSD and reinstalling MacOS onto that.

For more information see Use an external SSD as your startup disk … - Apple Community.

A late 2013 iMac cannot be upgraded beyond macOS Catalina 10.15.7. See the following from Identify your iMac model - Apple Support

iMac (21.5-inch, Late 2013)

Model Identifier: iMac14,1

Part Number: ME086xx/A, ME087xx/A

Newest compatible operating system: macOS Catalina

However, there is no reason that I know of that it should be "super slow" and that isn't an area that I have much knowledge. I would suggest that you create a new post in this forum with a title something like "Very slow iMac" and include the information that you have in this post. That may attract the attention of someone here who can help.

+1 to what padams35 wrote, an SSD will help greatly vs. a conventional 1TB 5400 rpm HDD… the RAM is upgradeable in that iMac up to 16 GB but not easily by the user as it involves diving into the “guts” of your iMac, which is more a job for your local AASP ( apple authorized service provider) or Genius Bar+ local Apple Store…

John B

How large is your HD and how full is it? I also see you have 8 gig RAM. These two things could be cause for system to slow down. Also, alot of chuff is generated over time. You might want to conside re-installing OS. Restart and immediately hold either R key or Command-R to boot into recovery mode. Run First Aid then choose Re-Install OS. This will overwrite the OS and create fresh system files. This also clears out alot of files that have piled up. Don't worry about causing any harm as long as you don't erase and re-install. You will see alot of hard drive space open up and things will be faster for awhile.

Hope this helps.

You may want to post an EtreCheck report and someone can tell you why your Mac may be slow.

click here ➜ Using EtreCheck - Apple Community

GaldoD wrote:

My iMac has become super slow over the years and as you are all aware, Apple doesn't offer support for older devices because they want you to throw it out and buy a new one

You apparently don't understand how the progression of technology works...

You realize that your Mac is 10 years old, right?

And that the original Mac, with a Motorola 68000 processor and 128 KB of RAM came out in 1984.

That's only 39 years. Your Mac just can't handle the advances in software that have been made in that time. It's completely unreasonable to expect it to do so.

"Super slow" can happen if you fill your startup drive almost to the brim.