I use toner transfer regularly. Luckily there's a printing press next door and I have a wide choice of paper types. The one I generally use is somewhat thinner than copier paper, glossy on one side and matte on the other. I don't know what it's called but it's not photo paper.
After the ironing, I soak it in water. Rubbing on some dish cleaning soap helps penetration. Then I rub off the paper with my thumb under a trickle of water - not too roughly and not too gently. After most of the paper is gone, there's still a fuzz of the fiber sticking out at the sides of the tracks ( not visible when wet). The fuzz does interfere with quick and neat etching, so I scrub off as much of it as possible with a toothbrush - again not too vigorously.
As someone else pointed out, your problem is with the paper coating. With the paper I use, I routinely use 20 mil (0.5 mm) tracks and occasionally do less than 10 mil, but I try to avoid anything thinner than 15 mil. That's for circuit tracks. For text and markings, 10 mil is not difficult.