If you need to reduce the size of a screenshot it's often better to keep it as a PNG and reduce the number of colors. 256 colors nearly always carries all the information needed without blurring the edges or the text. Often 128 or 64 is fine. Don't use dithering - it harms the compression ratio, so you may as well use a few more colors instead.

Often just applying lossless PNG optimisations using a tool like https://imageoptim.com/mac will sometimes save a large percentage, although it can take a minute or so for the tool to finish.