Yes, I am afraid you have done wrong. Beetroot are NOT one of the plants which you can transplant. They have to be set where they are to grow. They do not tolerate any root disturbance. You might be able to get away with setting the seed in a plug tray then planting out the plugs when the plants are around 3/4" high (max.).
The normal way is to set the seed where they are to grow, when the seedlings are quite small chop them "out with a hoe leaving groups of four or five the width of the hoe blade apart. Later, you single out the groups (pull out the weakest, leaving only the strongest one of the group).
What is normally thought of as a seed is actually a number of seeds clustered together, so having set one "seed" you get several plants close together.