Fixed Point Combinator Turing proof
Your second $\beta$ reduction is wrong.
After the first reduction, you (should) have $(\lambda x.x (U U x)) g$, so after the second you have $g (U U g)$ as required, but you haven't substituted the second $x$ correctly, leaving both it and the $g$ that the function is applied to, giving you $g (U U x) g$, though you have substituted the first $x$ correctly.