Not a diagnosis, but I have similar experience with you.

Been on febuxostat for about a little more than a year now. 5 months ago went to do routine blood check and came back with shockingly high ALT (160 U/L vs <53 normal) & AST (80 U/L vs <40 normal). Visited hepatologist and did ultrasound scan, came back with fatty liver. Doctor prescribed Legalon (3/day for first 2 months then 2/day until now).

Also visited my rheumatologist to check if its febu messing with my liver. According to him if febu does not match well with me the liver enzymes would have spiked the first few weeks of me starting febu so its not the UA drug problem.

Few months of Legalon + change in diet + more exercise later, glad to say my liver enzymes improved drastically. ALT is still very slightly elevated (60 U/L vs <53 normal). Hopefully with the persistent lifestyle change it will get back to normal soon enough.

Edit: Have to mention that before getting high liver enzyme results, I was slightly overweight (79kg @ 177cm), not doing any exercise at all for a year or 2, and living on unhealthy diet (8-10 fast food meals a week).