What am I doing wrong solving this system of equations?

Hint: Try inputing the solution $(3,2,1)$ into every step. That will allow you to identify the step where you went wrong.

You do (in the third matrix): $$L3-L4=(0, -3, 1 \mid -5)-(0, -13, 9 \mid -19)=(0, 10, -8 \mid 12)$$ but you have $(0, 10, 8 \mid -12)$ instead.

Note 1: Referring to 5xum's answer, in case you don't know the final answer, any solution will do, e.g.: $$(x_1,x_2,x_3)=(0,0,-1);(0,0,-1);(1,0,-1);(12,0,0) \ \ \text{(respectively)}$$ Note 2: In step $3$, you can reduce column $3$ instead of column $2$: $$\left[ \begin{array}{ccc|c} 1&8&-7&12\\ 0&2&-3&1\\ 0&-3&1&-5\\ 0&-13&9&-17\\ \end{array} \right] \Rightarrow \left[ \begin{array}{ccc|c} 1&8&-7&12\\ 0&-7&0&-14\\ 0&-3&1&-5\\ 0&14&0&28\\ \end{array} \right] \stackrel{\frac{-R_2}{7};\\ \frac{-3R_2}{7}+R_3}{=}\Rightarrow \left[ \begin{array}{ccc|c} 1&8&-7&12\\ 0&1&0&2\\ 0&0&1&1\\ 0&14&0&28\\ \end{array} \right]$$ The second and fourth equations are dependent, therefore they produce the same solution $x_2=2$. You can finish the problem now.

Note 3. When you can not find your mistake (sometimes the brain gets blocked/accustommed and can not see obvious mistakes), leave it for some time (1 hour, 1 day) and return with fresh mind. You will be surprised to easily spot the error.