Amending the Soil

We have a soil composed of a thin layer of topsoil over clay you could use for making pots. It is red. It holds together and can be molded when wet. It is the color of a flower pot. 

I tried growing carrots, radishes, and beets in the spring. The beets and radishes grew on top of the ground, and were small. The carrots came out deformed, even though I grew a stubby variety adapted to clay soil and dug the area extensively. 

I have finally broken down and asked my husband to acquire some sand. He brought home a few buckets-full for me, and has even offered to dig it into a bed. This entire area was once under a shallow sea, and is still well-watered, so everything under the topsoil is layered sand and clay.The layers can be many feet deep, though, so I can't just dig down to sand and mix to solve the problem. Hopefully adding sand now and organic material when I put the bed to rest in late November will help. I'm trying to save as much organic material as possible, to build soil so we can grow more stuff next year. We shall see.