When above-ground plants do well and root crops do poorly, it's generally due to heavily compacted soil. Imagine trying to push your little self through dense clay!
You can't expect to sow carrot seed in rock hard clay (especially clay that you've been further compacting with your big feet) and get anything other than some green growth up top and a small, unhappy and misshapen rear end underground. Carrots require the absolute loosest soil of pretty much anything we grow.
That's a raised bed (where the soil is never walked on) with excellent drainage at the very least. Even better would be a raised bed into which you have mixed a good amount of something to loosen and lighten the soil, like perlite (a mined, volcanic mineral that's popped into little balls that look like Styrofoam) or a soil-free mix designed for seed-starting.
Better than that would be a nice deep container with excellent drainage filled with (ideally) one-third soil-free mix, one-third perlite and one-third light, screened compost. This is not a place for any kind of {quote} 'topsoil' or garden soil even if your garden soil is naturally sandy. (But if that is the case, try growing some carrots in a raised bed of that sandy soil plus some compost.)
