The on-loan Grasshopper Zurich forward, whose deal ends in January, went off during his side's 4-2 Scottish Premiership defeat by Dundee United on Sunday.
Manager Derek Adams, whose team host Hearts on Saturday, said: "It's a big blow because Orhan is a strong player and is technically very good. We are going to miss him for about three to four weeks."
Adams did not speak to the media after the home loss and he revealed his frustration with some refereeing decisions, along with his side's defending and luck, meant he felt it was best not to talk.
United took the lead through a Gary Mackay-Steven penalty after goalkeeper Mark Brown was adjudged to have brought down the winger, and Adams felt referee Brian Colvin rejected better claims from his players Stuart Kettlewell and Kevin Luckassen.
Adams said: "Looking back on the stats, we had 60 per cent possession in the first half, had more shots on goal and crosses into the box and we were 3-0 down.
"That was difficult to take. Up to the 15-minute mark there was not too much in the game, we had the better of it, and they get a penalty that in my view was not a penalty.
"Mackay-Steven goes into Mark Brown rather than Mark Brown taking Mackay-Steven out. Mackay-Steven was probably looking for the penalty.
"The second and third goals, we could have done better defensively, and the fourth goal was offside, and they get a 4-0 lead.
"We hit the woodwork twice in the first half and had two penalty claims turned down ourselves. So on another day the game could have turned out differently if circumstances had been in our favour.
"The Kettlewell one could have been a penalty and the Luckassen one was definitely a penalty.
"The referee didn't make a decision on the Dundee United penalty until very late and he didn't get anything from his linesman."
- Sports & Recreation
- Dundee United
- Stuart Kettlewell