That doesn't sound like sign vinyl.. And if it's that thin,, the blade would have to reset for it...too much blade exposed. Then maybe it wouldn't curl up while cutting. Cheap sign vinyl is usually thick., stiff. At least what I have seen..
To start with, you should set your blade depth correctly, by taking the blade holder out of the machine, and firmly cut across a piece of scrap vinyl, you will be cutting. You should only be cutting the vinyl and barely a mark on wax paper backing, Adjust blade to get there, Then put the blade holder back in machine, and use the force of the machine to get there, same results, only cutting the vinyl and barely a mark in wax paper backing.