Growing carrots like the ones in my image below requires really good conditions, many of the top growers grow them in a Polyhouse/Tunnel in order to maintain a good warm temperature which make the carrots grow strong and even.
A good friend of mine Peter Clark, grows them in tall barrels filled with a John Innes No 2 mix of soil, the soil needs to be of a fine riddle mix so no lumps are present that causes the carrots to have bent roots.
Buying in John Innes No 2 is the best policy as you get clean sterilised quality soil from the start, if you were to try and make your own from garden soil it can takes many years to get the soil in the same condition that it comes out of the bag. Its the expensive way but its the guaranteed way of success.
The best long carrot to grow are the New Red Intermediated types , I have a good selection here
For the short carrots I have a new variety Abaco F.1 hybrid that was very impressive at the Vegetable Olympics held at Harrogate Autumn Show this year , another that wins many championship shows is the variety Sweet Candle F. 1 hybrid. Abaco is a very short true stump variety I introduced in 2009, It is a Chantenay X Kuroda type and is bred for its great colour and tender eating qualities.
Sweet Candle is a long Nantes type and again is excellent for the cooks among you.
Top quality new short carrot Abaco F.1 Hybrid