If you've built raised beds and put soil in there that's been moved, and even mixed in some peat, then the soil
should be loose enough for a carrot to get through. Carrots don't like
root disturbance, so avoid working right near the
roots. Don't fertilize your carrots, that makes them grow more greens than carrot. If the tops aren't growing for a bit but the roots are (the carrots) that could actually be a great success, you might be doing just fine. I gauge root diameter occasionally by sticking a pinkie in the soil on either side and seeing how thick it is. If that's moving forward, you're doing great. Also don't forget to thin them out, make sure there's only one carrot root growing at each location.