How do I get rid of this dirt between the tiles of my bathroom shower?
I am in the US.  I have had luck with Soft Scrub (bleach and mild abrasive in a paste like base), and with Zud (oxalic acid); Bar Keepers Friend is also oxalic acid.  Comet is also bleach.  Get one of those products and Scotch Brite green scrub pads or something similar.  Scrub the stuff in there and leave it there.  Let it sit all day before you rinse it off.  

Plain bleach is really less than ideal for this for a number of reasons.  One it does not cut grease and grime at all.  Two it is runny and won't stay on a vertical surface.  Three you will breathe a lot of it and hurt your eyes and nose.