I noticed that my toilet flushes poorly, that is, the rinse water flows in at top left, and circles the bowl, but does not do a good job rinsing down the back of the bowl. This quickly lead to an odor problem, although I did not notice yellow water on top of the flush flap. Perhaps that is what is happening to yours?
Anyway, our initial solution was a 1 liter soda bottle full of water left in the shower stall, and when you finished your business, you poured a quick rinse from the soda bottle at the back of the bowl as you flushed.
That was a bit of a pain, but a cheap fix. Then I discovered a $50 hand bidet at Lowes. It rather looks like a kitchen sink hand sprayer, but it taps into the water line to the toilet and has a handy screw holder thing that goes on the back of the shower wall, so it looks like it belongs there. Being right beside the toilet, it's a quick squirt in the bowl, and beautifully, it works great as a bidet, too. No more soda bottle, professional-looking installation, cheap, double-purpose, what's not to like?
Oh, it is also really handy for rinsing down the shower stall floor or walls. Much easier to use than the shower head.