O'Hara holed four birdies in a bogey-free second round, putting the Scot 11 under heading into the weekend at Serengeti Golf Club. He needs a top-four finish to save his European Tour card for next season.
Goosen, a two-time champion at this tournament, showed signs he has shrugged off his nagging back problems with five birdies and a bogey en route to a 68. Bremner also had a 68 thanks to birdies on two of his last four holes.
Defending champion Ernie Els shot a second straight 69 for a share of 14th place at 6 under.