There are two ways to use a 4K monitor on a Mac. The first way - the one you appear to have tried so far is to run it in a native (for want of a better term, Apple call this Default for Display) resolution whereby you set a resolution e.g. 4K and this is the resolution the software sees and this gives you 4K of pixels and means a letter that uses say 12 pixels will be very, very small. In this 'native' mode, setting the monitor to 1920x1080 mode means that you get bigger text but do not benefit from the extra 4K resolution to get better quality text.
The other mode which Apple call Scaled 'virtualises' the resolution. With this approach you can use a slider to adjust how big the text appears and behind the scenes it uses the full 4K of pixels to draw it so this gives you the best of both worlds. You can set it to bigger text and still get very smoothly drawn 4K resolution text. As a brief explanation of how this works the image is drawn in memory at a very high resolution and then shrunk to fit on the screen, this is also how the iPhone and iPad work.
So, try opening System Preferences, then click on Displays, then change the mode to 'Scaled' and try adjusting the slider so you get the size of text you want.
Note: While the text can be made bigger this way and still keep the 4K pixel quality making it what Apple call a 'retina' type display photos etc. should be drawn in full resolution making them look far better quality than when you set the monitor to 1920x1080.