Five-times European Tour winner Sterne, who carded a two-under-par 70 for a halfway total of 132 (12-under), found the going a great deal tougher in the swirling winds than when he opened with 10 birdies in a flawless round on Thursday.
Former world number one Westwood was also one of those to struggle in the testing conditions later in the day, the Englishman left to rue a double-bogey at the 17th in his 71.
Young Dane Thorbjorn Olesen (66) equalled the low round of the day to sit level with England's Tommy Fleetwood (68) and Scot Stephen Gallacher (70), one behind Sterne.
Scotland's Chris Doak (69) led three players on 10-under while former world number two Sergio Garcia, who suffered with a shoulder injury during Friday's round and thought about pulling out, was among three players on nine-under.
World number 165 Sterne said acclimatising to the different conditions had been his toughest test.
"The course was a lot tougher with the wind and the greens definitely firmed out a bit. I think it's going to be quite tricky coming to the weekend," Sterne said.
"There's a lot of guys bunched up and you're going to have to play pretty well to be there on Sunday," added Sterne, whose last win was in December 2008 when he claimed back-to-back titles.
Olesen, who claimed his first tour win last year and will make his U.S. Masters debut in April, has played 36 holes in Dubai without a bogey and said his confidence was sky high.
"I believe in myself and want to achieve my own goals - that's the biggest pressure. I'm hungry for more wins. I want to win twice this season. That's my main goal, so the quicker I can get the first one the better," said the 23-year-old.