Edges visible in the middle of a face

I’m trying to model this coffee table (http://www.archiproducts.com/fr/produits/155197/maze-table-basse-en-plastique-maze-side-table-quinzeandmilan.html):

I started with a plane and extruded vertices to model the feet.
After getting the overall shape, I added a solidify modifier, and a bevel. But the result is awful, some edges inside faces are very pronounced…

I tried to add vertices to form quad, but it seems to be worse as the edge are more pronounced…

How can I prevent that ?

Here is the blend file if you want to see the disaster…
maze-side-table.blend (421 KB)

Thanks for your help.