The dual Dubai World Cup winner was racing over an inadequate mile on his first outing since defending his crown at Meydan.
Ridden as usual by Christophe Soumillon, he had no trouble laying up with the pace and made a bid for home inside the final furlong only to flatten out inside the last 100 yards to be beaten three-quarters of a length and a neck by Mitole and McKinzie.
He will now be prepared back in Newmarket before returning for a race at Saratoga over a furlong further in August.
“Thunder Snow race a huge race and I am happy with him,” Bin Suroor told www.godolphin.com.
“He was up against some of the best American milers and will improve for the race.
“We will bring him back to England and prepare him to race in America again, with his target being the Grade One Whitney Stakes (August 3) at Saratoga.”