Horschel, who set a USGA scoring record with an 11-under 60 in the first round of stroke play, defeated fellow American David Merkow 3 and 1 to advance to the third round Friday.
Afterward, Horschel admitted his newfound status among the competitors may have its downside.
'There's a bull's eye on me,' Horschel claimed. 'Some may say there isn't, but I'm the one who shot 60. I'm the medalist, so everyone wants to gun for the top guy. Someone's lying if they say they're not gunning for the top person.'
Tway, the 2005 U.S. Junior Amateur champion, failed to win a hole in his 6 and 5 loss to Canadian Ryan Yip, who moved on to face Horschel in the third round.
Yip downplayed questions about his matchup with the toast of the tournament thus far.
'I don't really know who he (Horschel) is,' Yip admitted. 'I only know he shot 60. So, he's a great player. Hopefully, I have no idea what to expect. I won't worry what he's doing because I can only control myself.'
Play was delayed for nearly 4 1/2 hours in the morning because of dangerous inclement weather. Consequently, only one of the two scheduled rounds was played, forcing organizers to amend the schedule to include third round and quarterfinal round matches on Friday.
Fourteen other players advanced to the third round, including a familiar face who removed a rather large monkey from his back.
Trip Kuehne, the brother of 1998 U.S. Amateur champion Hank Kuehne and three- time USGA champion Kelli Kuehne, moved on with a 2 and 1 victory over Casey Watabu.
Since losing to Tiger Woods in the 1994 final, the elder Kuehne, 34, had qualified for match play four times, only to lose in the second round each time.
'I'm relieved,' admitted Kuehne, who led for the final 10 holes of his match and closed it out with a par at the 17th.
The other winners Thursday were Oliver Fisher, Jonathan Hodge, Matthew Swan, John Kelly, Jon McLean, Roberto Castro, Webb Simpson, Alex Prugh, Mark Strickland, Kyle Davis, Richie Ramsay, Rickie Fowler and Pablo Martin.