The power available is flows times drop. If a 4" pipe is always full, then you could get more flow with a 6" even if it isn't full.
The pounds/square inch pressure at the bottom of the pipe is the same regardless of pipe diameter, but the total pressure will be greater with a larger diameter pipe (feet of head is feet of head).
Go with 6" all the way until you get to your turbine, then install whatever adapter is needed. If you restrict the flow at the turbine to match the flow from the spring, the water will back up to the top of the pipe drop and you will get the max head at the max flow.
A 6" pipe will pass 2.25 times as much water as a 4" pipe (all other things being equal.