Anytime water quality changes after a rainstorm, it is not a good thing. It usually means the well is not sealed properly, and rainwater is getting in. Believe it or not, rainwater can contaminate the water in the well. It flushes pesticides, dog crap, and anything laying on the ground down the well. So first thing, the well needs to be sealed properly.
Running the sprinkler for a long period of time can draw the water down in the well, causing high velocities, which can stir up the well.
Or running one sprinkler for long periods can stir up the well by cycling the pump on and off. Cycling can also stir up the well and cause problems.
After you get the rainwater intrusion stopped, you can pump the well hard and see if you can get it developed and cleared up.