Your question is a bit misleading, you state: The static pressure measured with the outside faucet is 60 psi. When irrigation system is turned on, the water coming out of all sprinkler heads in any zone is weak." I am assuming the so PSIG drops with the sprinklers, if that assumption is wrong you have a restriction or partly closed valve in the sprinkler feed.
I have city water and the sprinklers did the same thing, I added a length of 1/2" copper tubing in the feed and it now works great and the inside pressure is good.
I am assuming the pump does not shut off, this is the way I would expect it to operate. The pump can pump so much water at a given pressure. Use more water the pressure will drop, use less and the pressure will go up.
Many sprinklers will allow you to adjust the flow. If you have that type of sprinkler head lower the flow and let them run longer. The end result the area gets the same amount of water in the end. That would be by far the most cost effective. As mentioned earlier you can add more zones.