Electronics

[last mango]

You don't have to delete on each device.  
One of the previous posters gave you instructions to turn iCloud on for your devices.
Follow that and get it set up 😀  You need to have iCloud turned on for all your devices and in iCloud have iMessage turned on. 
 
I have an iPad Pro, an iPhone 11 and an iMac and when I delete a text on any one of these devices it deletes it on all of them since they are all synced up with iCloud.  

Now I think I need to add that this of course only works if the text you are deleting comes from an apple device ( iMessages are blue).   That is how it all syncs up.  If the person sending you the text has an Android (the message will be green) then you will have to delete on each device.   This could be why you are having to delete on each device.  

Hope this helps 😀

[Cats3000]

I can delete text messages at will on my iPhone.  I can't delete them at all on my iMac.  They eventually do drop off.