O’Hara holed four birdies in a bogey-free second round of 4-under 68 for the Scot to head into the weekend on 11 under at the Serengeti Golf Club. He needs a top-four finish at the tournament to save his European Tour card next season.
Goosen showed signs he has shrugged off his niggling back problem with five birdies and just one bogey as the two-time champion hit a 68. Bremner had the same score after he birdied three of his last five holes to move to 10 under alongside Goosen.
Defending champion Ernie Els shot a second successive 69 for a share of 14th place at 6 under.
O’Hara, 31, kept his nerve in windy conditions and on fast, testing greens – and with his immediate tour future in the balance – to retain the lead he shared with South Africa’s Jbe Kruger on Thursday.
He had birdies at Nos. 2, 11, 16 and 17 for a one-stroke advantage ahead of the biggest weekend of his year.
“You look at the guys behind me like Retief Goosen, they can score really low and I’ve still got 36 holes left,” O’Hara said. “I’ll keep doing what I’m doing and trusting my swing and if I’m in a position to win on Sunday then so be it. Otherwise, hopefully I’ll get a top three or four finish.”
Runner-up to Els in 2010, Goosen birdied his first two holes but dropped a shot at the par-5 No. 4. He responded with another three birdies, including one on the last where he coolly rolled in his putt for a 10-under two-round total of 134.
Having spent months struggling with back problems, Goosen is in contention for a first tournament win since early 2009.
“I can only see the greens getting tougher and that will make the scoring harder,” he said. “But the tougher it gets the better. When it becomes a grind, that’s the way I like it.”
Bremner jumped into a tie for second after a tumultuous finish at the Jack Nicklaus-designed Serengeti, where he followed a bogey at No. 12 with a double-bogey six at No. 13 and then birdies at Nos. 14, 15 and 18.
South African duo Garth Mulroy and Lyle Rowe are two strokes off the lead in a tie for 4th on 9 under. Rowe was the day’s biggest mover with his 6-under 66, which included an eagle and six birdies as well as two dropped shots.
Els had two birdies, then two bogeys, then two more birdies in an up-and-down opening six holes. The five-time South African Open winner had just one more birdie and eight pars coming home to lie five shots off the lead.
Denmark’s Thomas Norret (69), Frenchman Jean-Baptiste Gonnet (70), English pair Simon Wakefield (67) andGraeme Storm (68), and former Ryder Cup player Phillip Price of Wales (70) were all alongside Els at 6 under.