Advice on shower stall leakage

CC
We've moved into a great family home. Just one issue. We have two showers, and both shower cavities are level with the bathroom floor. This means that every time we shower water seeps out beyond the glass door onto the floor surrounding the shower where we step out.
 
Currently, we keep our bathmat beyond that so it doesn't get soaked, but do have to use an extra towel or bathmat constantly to mop it up. It then dries out but often smells of damp. Not to mention, super frustrating!
 
Can you or your husband / boyfriend offer advice? It is a rented home, so we can't do anything major. Although we could ask permission if need be.
 
Photos (not too bad here, but this is hours after showering):

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