It is allowed, as a US citizen you can enter the Schengen area for up to 90 days in any 180 day period and this plan would take 4 days (each calendar day is taken as a whole day, even when you leave just after midnight or arrive just before midnight.)
But I think you are taking huge risks, with the number of flights cancelled or people being booked on an earlier or later flight.
If you want to do nested flights, have a couple of days between the arrival on an airport and leaving from it and even then you do run a risk if the flight is just cancelled and the airline can not re-book you in your window.
You can get stuck in Paris with no way to get to Istanbul, or in Istanbul with no way to get to Amsterdam. Or indeed miss your flight from Amsterdam as your flight from Istanbul is delayed (or re-booked) by a day or your airline re-books you a day early from Amsterdam.
If you book separate tickets, as this plan seems to be about, you are left on your own if any of the flights is not working out for you, often making it needed to buy a new ticket against the prices of the day (often very expensive.)
I think you are better off with a single booking (made up to be one ticket by a reputable airline) which includes all 4 tickets. In that case the airline will get you to your final destination (including home) if any of the flights is cancelled or the time changed to make the connection impossible.
Intercontinental flights are not things to go for the cheapest option these days, much to much risk of things going wrong.
Yes, the flights all on one ticket are more expensive, that is the insurance part of the ticket.
If you had a month in Europe and out of that you wanted to go a week to Istanbul, I would say go for it, but not with just a one day between flights.
All above is without the added risks of your Turkey travel plans, where also a lot can go wrong.
And that is also besides the fact that you will almost spend more time waiting in airports than walking in the cities you want to visit, better have 4 days in Paris and 3 in Amsterdam or fly direct to Turkey and have half your time in Istanbul and the rest in two (or even one) smaller town or some countryside or coastal destination.
Added after the change of the question, (flights to and from Turkey are already booked as a single ticket.)
You do not run the risk of getting stranded on your Schengen stops, as your airline ticket will cover you there.
But you do still run the risk of being stuck somewhere in Turkey and you will still spend, relatively, a lot of time in airports waiting for flights.
I have personally noted that a flight of an hour will eat up at least half a day with getting ready and getting to the airport, checking in early enough to be sure to make your flight, and after your flight getting into the town and getting to your next hotel.
My personal advice is to base yourself in one place and see and do things around there, much less time spend in transit so much more time to see things. (It forces you to a different speed, you may see fewer different areas but you will see that one area much better, and have time to actually see things.)