I had 4 punctures happen in a similar fashion: they all punctured with an explosive "whoosh" sound a few minutes after inflating. Here are pictures of punctures 2 to 4.
As you can see, they look almost identical. All of them are facing the sidewall of the tire, but they are in different places.
After the 1st puncture, I thought that there could be debris in the tire that caused it, so I thoroughly felt inside the tire, around the tube, and inside the rim. There was no debris and I did not feel anything sharp. I patched it, reinflated just the tube to make sure I fixed it, then fitted it back on the wheel.
After the 2nd puncture, I bought a new inner tube, thinking the last one was bad. Same thing happened. Admittedly, I did not have the best technique for fitting the tire on because I was sort of sweeping the lever around the entire wheel, so for the 4th attempt, I used my hands as much as possible until the very end to lever the last bit in. Additionally, I fitted the tire from the other side with the cogs facing me, and I remembered the location where I used the lever. I squeezed the tire all the way around, making sure that the inner tube was not pinched.
As this picture below shows, the 4th puncture happened on the same side as the 3rd, which leads me to believe that my technique is not the cause. The puncture is nowhere near the area where I used the lever. I checked for debris or sharp edges again carefully, and there is nothing.
I've had this bike, wheel and tire for 6 years, and this sort of puncture has never happened before. What would you do at this point? I could replace the tire to a Gatorskin. But nothing about the tire or wheel changed when this started happening. Another option is to go tubeless but I don't want to put in the work required for it.
Edit: I realized that there was a hole in the tire where the 4th puncture occurred. When the puncture happened, I felt a puff of air coming out of the tire. I didn't think much of it then, but could this be a big hint as to what happened?