I am not doing that much weights so I don't know what a supinated grip is.
But I DO know that in my sports (aerobox and rowing) people sometimes feels alot of pain in the forearms after training and have troubbles lifting/gripping anything at all. even the smallest pencil or fork can be difficult in the worst cases. anyway, what's recommended then is often to stretch the forearm by keeping the sore arm straight, elbow pointing towards the floor (palm towards the ceiling) and get hold of the fingers with the other hand, and stretching them also towards the floor for 10-20 sec, rest and then stretch again. Repeat this set as often as you can in your everyday life - while waiting for the bus, at the movies, infront of the tv - for 3-4 weeks. Do at least 8-10 sets a day.
doing dishes by hand in really warm water is another thing that some people say can be helpful with sore forearm situations, it's like watergymnastics for the wrists and forearms. and the dishes are being done at the same time, isn't that practical
I'm not sure if these tips will help your lifting situation (as I said, I don't know anything about weightlifting) but they usually help people with pain in their forearms due to gripping and pulling stuff on a regular basis
Bronco - If I didn't could make my self understood in english about the stretching move, just let me know and I'll explain it to you in swedish