I was having problems for months with my MBPro Mid-2012 15" (first Retina model) and external Japanese monitor (21.5"). Fan blasting constantly, lagging, browsers locking up. Tried everything, including upgrading from High Sierra to Catalina but no go.
The one tip that resolved it all was setting the resolutions for both monitors exactly the same. I couldn't do that through System Preferences/Displays/Scaled option. I used a third-party app to precisely select 1080p (in my case) for both displays and voila, fan shut off, no lagging, all fine.
Why it wasn't an issue in the past with the same monitor I have no idea, but once the problems started nothing would resolve them. Hope it helps.