There is evidence for a link between fructose consumption at excessive levels and markers of non-alcoholic fatty liver disease, such as increased liver fat storage, but not for consumption of sugars at normal dietary levels. These findings seem to be attributable to increased energy intake from fructose, rather than a unique property of fructose.

Key research 

Chiu S, Sievenpiper JL, de Souza RJ, et al. (2014). Effect of fructose on markers of non-alcoholic fatty liver disease (NAFLD): a systematic review and meta-analysis of controlled feeding trials. Eur J Clin Nutr, 68(4):416-23.
Isocaloric exchange of fructose for other carbohydrates does not induce NAFLD changes in healthy participants. Fructose providing excess energy at extreme doses, however, does raise intrahepatocellular lipids (IHCL) and alanine aminotransferase (ALT), an effect that may be more attributable to excess energy than fructose.

Other research

Jensen T, Abdelmalek MF, Sullivan S. (2018). Fructose and sugar: A major mediator of non-alcoholic fatty liver disease. J Hepatol, 68(5):1063-1075.
Early clinical studies suggest that reducing sugary beverages and total fructose intake, especially from added sugars, may have a significant benefit on reducing hepatic fat accumulation. The authors suggest larger, more definitive trials to determine if lowering sugar/HFCS intake, and/or blocking uric acid generation, may help reduce NAFLD and its downstream complications of cirrhosis and chronic liver disease.

Schwimmer JB, Ugalde-Nicalo P, Welsh JA, et al. (2019). Effect of a low free sugar diet vs usual diet on non-alcoholic fatty liver disease in adolescent boys: a randomised clinical trial. JAMA, 321(3):256-265.
In this small short-term trial in teenage boys with non-alcoholic fatty liver disease, a diet low in free sugars compared with usual diet resulted in a greater reduction in hepatic steatosis from 25% to 17% in the low free sugar diet group and from 21% to 20% in the usual diet group

Bravo S, Lowndes J, Sinnett, S, et al. (2013). Consumption of sucrose and high-fructose corn syrup does not increase liver fat or ectopic fat deposition in muscles. Appl Physiol Nutr Metab, 38(6):681-8.
When high-fructose corn syrup (HFCS) or sucrose were consumed at normal levels, there was no change in the fat content of the liver, vastus lateralis muscle or gluteus maximus muscle. The data suggests that when fructose is consumed as part of a typical diet in normally consumed sweeteners, such as sucrose or HFCS, ectopic fat storage in the liver or muscles is not promoted.

Luukkonen PK, Sädevirta S, Zhou Y. et al. (2018). Saturated fat is more metabolically harmful for the human liver than unsaturated fat or simple sugars. Diabetes Care, 41(8):1732-1739.
Overfeeding saturated fat increased intrahepatic triglycerides more than unsaturated fat. Simple sugars increased intrahepatic triglycerides by stimulating hepatic de novo lipogenesis. Saturated fat significantly increased lipolysis while unsaturated fat decreased lipolysis. Saturated fat induced the greatest increase in intrahepatic triglyceride, insulin resistance, and harmful ceramides.

Chung M, Ma J, Patel K., et al. (2014). Fructose, high-fructose corn syrup, sucrose, and nonalcoholic fatty liver disease or indexes of liver health: A systematic review and meta-analysis. Am J Clin Nutr, 100(3):833-49.
The apparent association between indexes of liver health (i.e, liver fat, hepatic de novo lipogenesis, alanine aminotransferase, AST, and γ-glutamyl transpeptase) and fructose or sucrose intake appear to be confounded by excessive energy intake. Overall, the available evidence is not sufficiently robust to draw conclusions regarding effects of fructose, HFCS, or sucrose consumption on NAFLD.

Jia Q, Xia Y, Zhang Q, et al. (2015). Dietary patterns are associated with prevalence of fatty liver disease in adults. Eur J Clin Nutr, 69(8):914-21.
High-carbohydrate/sweet pattern scores are associated with higher prevalence of non-alcoholic fatty liver disease in females.

Johnston RD, Stephenson MC, Crossland H, et al. (2013). No difference between high-fructose and high-glucose diets on liver triacylglycerol or biochemistry in healthy overweight men. Gastroenterology, 145(5):1016-1025 e2.
In the isocaloric period, overweight men who were on a high-fructose or a high-glucose diet did not develop any significant changes in hepatic triacylglycerol or serum liver enzymes. However, in the hypercaloric period, both high-fructose and high-glucose diets produced significant increases in these parameters, without any significant difference between the 2 groups. This indicates an energy-mediated, rather than a specific macronutrient-mediated, effect. 

Schwarz J-M, Noworolski SM, Wen MJ, et al. (2015). Effect of a high-fructose weight-maintaining diet on lipogenesis and liver fat. The Journal of Clinical Endocrinology & Metabolism, 100(6), 2434-42.
Short-term high-fructose intake was associated with increased de novo lipogenesis and liver fat in healthy men fed weight-maintaining diets.