I too have found that reading skill books, has not been implemented properly in sims 4. Your sim can progress to level 10, by just reading level 1 books, making level 2 and 3 pointless. I get my sim to complete reading a skill book, by turning autonomy off, and disabling the decay of household needs, so that the sim reads continuously without a break. However, if the sim takes a break, then the skill level does not progress until the next level book is selected. So, If you cannot progress beyond a certain level, try the next level up and it should work.
