As thekeyclan says, the key bit is 90 days in the previous 180
So yes, you could have overlapping visas - you might have one from Jan 1 - June 30, for simplicity's sake, and only spend 60 days in Turkey in that time.
But your final trip is from June 15 to August 16, so you'd need a second visa to cover your time here from July 1 to departure.
But when you leave on August 16, you can't have been in Turkey for more than 90 days, counting backwards 180 days (would be to around Feb 18, I think).
At least, that's how I understand it....