Can I sue a landscaping company for turning off the sprinkler from my new lawn but not turning it back on and now it’s dying?

You can sue the landscaping company for negligence.  Negligence is the failure to exercise due care (that degree of care in this case that a reasonable landscaping company would have exercised under the same or similar circumstances to prevent foreseeable harm).

In order to prove negligence, you will need to prove duty of due care mentioned above, breach of duty (failure to exercise due care), actual cause, proximate cause, and damages.

Actual cause means but for the landscaping company not turning on the sprinkler, would your lawn have died?  If the answer is no, actual cause has been established.  Proximate cause means were there any unforeseeable, intervening acts which would relieve the landscaping company of liability?  If the answer is no, proximate cause has been established.  Damages means the amount of compensation you are seeking in your lawsuit for negligence.  Your damages would be the cost of replacing your lawn.  You will need to mitigate (minimize) damages by selecting a replacement lawn that is comparable to the lawn that was lost.  If you were to replace the lawn with some expensive exotic garden, you would have failed to mitigate damages and your damages would be reduced accordingly.

Prior to filing a lawsuit, it may be possible to settle the case with the landscaping company's insurance carrier.  If the case is settled with the landscaping company's insurance carrier, NO lawsuit is filed.  If the case is NOT settled with the landscaping company's insurance carrier, you will need to file your lawsuit for negligence prior to the expiration of the applicable statute of limitations or you will have lost your rights forever in the matter.