A life captured in photos.
Your soil is too dense/compacted. You need to till deep, then add some vermiculite a lot of peat-moss. This will make the soil much looser and allow the carrots to expand down and out.
Root veggies WANT to be ignored, but they also need plenty of water and room to grow.
We missed out on watering...a lot. We didn't give our annuals or carrots any hope of surviving this year but next year I'll follow your advice and get carrots that are more than a finger long.