Jamieson followed his first-round 67 with a 65 on Friday, while Bjork has gone 65, 67 to leave both on 10 under.
Home favourite Thomas Aiken bagged the round of the day, a nine-under-par 62 that took him to eight under overall.
Glasgow-born Jamieson, 33, said: "It was very solid again from tee to green, like Thursday."
Jamieson has spent the last two seasons battling to retain his playing privileges, finishing 106th and 107th in the Race to Dubai, but won his sole European Tour title in South Africa at the rain-shortened Nelson Mandela Championship in 2013.
"I'm delighted with that today," the Scot added after his 65 at Pretoria Country Club. "I struggled a bit last week from tee to green but it seems to have clicked this week. Hopefully I can hole a few more putts this weekend.
"The rough out there is pretty extreme but there's not a great deal of drivers (off the tee), which is probably suiting me right now because I struggled last week with my driver."
England's James Morrison is a shot off the lead on nine under thanks to a second round of 66, with Scots Marc Warren, Richie Ramsay and Duncan Stewart in a share of 18th on four under thanks to second rounds of 67, 70 and 72. Fellow Scot David Drysdale is back on two under after a second round 68.
Scotland's Scott Jamieson shares the lead with Sweden's Alexander Bjork at the halfway stage of the Tshwane Open in South Africa.