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.

Boyle struck the only goal in the 18th minute after lobbing over Graham Doyle following a moment of hesitancy in the Wexford defence.
Derry needed fine saves from Gerard Doherty to stay ahead as the keeper denied Conor O'Keeffe and Paul Murphy.
Josh Daniels also headed a Gary Delaney effort off the line in the second half as Derry endured some nervous moments.
Derry did go close to adding to their lead in the final five minutes of the game as Aaron McEneff clipped the Wexford crossbar with a long-range shot and substitute Jordan Allan fired just wide.
Kenny Shiels' side suffered a blow in the second half as goalscorer Boyle was forced off by an apparent hamstring injury.
With leaders Dundalk beating St Patrick's Athletics 2-0, the Candystripes remain nine points off the pace but they have edged to within six points of second-placed Cork City who were held to a 0-0 draw by Shamrock Rovers in Dublin.

Environment body Natural Resources Wales (NRW) said staff were erecting temporary flood barriers and carrying out other prevention work.
An amber "be prepared" warning for persistent rain is in place for Gwynedd, Conwy and Denbighshire.
The Met Office said up to 15cm (6ins) of rain could fall in north Wales.
A yellow "be ready" warning is also in place for Ceredigion, Conwy, Denbighshire, Flintshire, Gwynedd, Anglesey, Powys and Wrexham.
NRW  has issued river flood alerts due to rising levels caused by continuous rain since Friday.
It said people should be prepared for flooding and it was monitoring the Dyfi, Elwy, Conwy and Vyrnwy river catchments.
Scott Squires, of NRW, said: "Our staff are out working putting measures in place to prepare for flooding. This includes erecting temporary flood barriers to protect areas at risk in St Asaph and Llanrwst."
Scotland's Scott Jamieson shares the lead with Sweden's Alexander Bjork at the halfway stage of the Tshwane Open in South Africa.
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Nathan Boyle's first-half goal proved enough to give Derry City victory over Wexford Youths at the Brandywell.
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Work is being carried out to try to prevent flooding in parts of north Wales as heavy rainfall is forecast to continue into Sunday.