Possible reasons for very laggy video on external 4K monitor from MacBook Pro (Retina, 15-inch, Late 2013)?
It's about the decoding codec. Chrome does not support hardware accelerated vp9 codec on Mac OS, which is required to decode any 4k video on Youtube. Which means when decoding the 4k video it's using your CPU's resource instead of the GPU. Good luck trying to decode any 4k video using a mobile CPU, it simply won't work, a 9900k might do it, but a mobile chip, nope. For safari it should support hardware VP9 decoding, however, your Macbook is so old I highly suspect that the GPU does not support VP9 decoding.
Anyway my Macbook Air 2020 has difficulty running 4k on Chrome, and always drops frame during the first few seconds of full screen on a 4k external monitor when using Safari. Time to buy a new laptop if you want smooth 4k video experience.