Not so much the weight of clothes as low-quality construction/materials.
Possibly compounded by "self assembly" if self assembly is not done with care - the drawer bottom should be fully seated in that groove.
Making new bottoms out of actual plywood as opposed to whatever Ikea used for the drawer bottoms is a solution. It's relatively costly and equipment/skill-dependent.
Solutions that may be less work and expense are to attempt to disassemble and reassemble "better or tighter" - potentially with glue in the slot (which will interfere with any future disassembly) and/or to glue small strips of wood to the bottom and sides of the drawer to hold the bottom in place better.
If trying that, read the instructions (available from Ikea's website) with care - it's always possible that the original assembly was incorrect in some detail. Steps 24-27, mostly. The little wedge seen in your blurry last picture is supposed to be holding the bottom in the slot.
At the low end of that would be just getting the bottom into the slot and then running a bead of hot-melt glue along the seam, which is potentially removable if you ever want to take it apart again, and does not require any wood strips. Using wood strips is a bit more secure, perhaps.