I found a solution to the fan running with an external monitor (2019 16" MBP, LG 4K).
I use an app called Lunar to turn off my Mac screen when I plug in the monitor. https://lunar.fyi/ It turns the screen fully off (in 'blackout mode'), not by just turning down the brightness. For me, that's enough to keep the fan off.
It has the same effect as putting the Mac in 'clamshell' mode (which I've heard also solves the fan problem), except this way you can still use the laptop's keyboard, trackpad, Touch Bar and speakers instead of having to buy externals.
It's expensive (20 euros), but there's a free trial to see if it works for you. For me, it was such a relief not having the fan on all the time.
I use an app called Lunar to turn off my Mac screen when I plug in the monitor. https://lunar.fyi/ It turns the screen fully off (in 'blackout mode'), not by just turning down the brightness. For me, that's enough to keep the fan off.
It has the same effect as putting the Mac in 'clamshell' mode (which I've heard also solves the fan problem), except this way you can still use the laptop's keyboard, trackpad, Touch Bar and speakers instead of having to buy externals.
It's expensive (20 euros), but there's a free trial to see if it works for you. For me, it was such a relief not having the fan on all the time.