With the current functionality, it is possible, but I don't think you're going to like how!
What you would need is for your network junctions to look like this:
Basically turns become a separate network line/link/edge.
Black circles are nodes and the red lines are the main network. You would need separate features for the different directions of travel (which is why there are two sets).
The blue lines are right-hand turns. The green lines are left-hand turns. I'm assuming it's a country that drives on the right.
All features would have a forward weighting of 1, and a reverse waiting of 9999. Where a turn was not permitted you would change its forward waiting to 9999 too.
Because you would be using forward and reverse weighting, the lines would need to be created pointing in the direction of travel.
With a road network like that, I believe you could use the ShortestPathFinder to get the result you need.
Of course, your data probably doesn't look like that, and would be a pain to modify. But other than that, I don't see how you can force that transformer to ignore a certain turn. There just aren't the parameters in that transformer.
With a regular network you could probably get FME to generate a route, test it for bad turns, adjust the network to avoid that, then generate a new route; a sort of iterative process. But it wouldn't be easy either.
I don't know if that will help any, other than to demonstrate how hard this would be. But I'll file an enhancement request too with the development team, and see if they can implement a solution (it wouldn't be soon).
And you should probably contact our support team (http://safe.com/support) too; firstly to get your contact info on the enhancement request, but also to see if they have any simpler ideas than me. Maybe I'm making this more complex than it needs to be!