Carrots and beets are some of the vegetables that are easy to kill with kindness. They’re little gluttons for space and nutrients, and must be handled with an iron fist to make them grow straight and strong. Give the buggers no slack at all! Your motto should be – “If in doubt, yank it out!” I pinch out a finger full (maybe 3/4” wide) and skip a finger width. Pinch and skip, pinch and skip, working with existing gaps and rooting out particularly thick clumps.
Raised beds may not be right for everyone, and our way is not the only way. I have seen raised beds made from rows of 5’ diameter kiddy pools, and heard of a fellow who collected junk refrigerators from the dump and lined them up on their backs into a rainbow of colored enameled steel raised beds. Even rows of five-gallon pails filled with plants count as raised beds in my estimation. Do it any way you care to, but do it if it’s right for you.