Can I make the seams between the tiles in my shower water resistant?
Tile, grout, and cement products are usually porous and will leak. The standard practice is to place a waterproof layer under the tile in showers and baths. Sealing grout is usually done for stain resistance, not to create a waterproof surface in a shower.

At this point, there's no easy answer. You'd have to fix it by removing the tile and starting all over.