As Dan says (of course
) plus: for future years, don't use fresh manure. You can prepare 'trenches' to grow your carrots
in by digging out a 'V'shaped trough the depth your carrots
claim to grow. Fill the trench with multipurpose compost and sow your carrots
into that. If you also loosen up the soil below the level of the bottom of the trench then any longer roots won't be stunted.
If any of your crop has carrot
rootfly maggots in, then you need to protect your crop next year, either with enviromesh for the full life of the crop, or by growing in a container up on a bench or inside a polytunnel.
Another possible reason for all the little roots could be that, in spite of regular watering, you have only been wetting the surface, so the carrot
has to grow lots of small roots to collect any moisture going. It can take a surprising amount of water to wet dry soil deeply. To check, you can either use a moisture meter, or dig a few test holes near the row to see how far down the moisture goes.