The garden is full of interlocked roots just under the surface. What can I do about it?

If they are tree roots you can end up killing the tree. I would add no more than 4 inches of top soil and plant in that for a couple years. You could build a wood frame work around the area so the dirt doesn't get washed away. Then if no trees died off add another layer of top soil. Continue over the years until you get it to the level you desire. Now, if you go with the whole amount you can end up killing anything that depends on those roots. If possible maybe you could follow one of them back to it base.