Flying through Italy from US to Turkey on non linked flights

Shouldn’t be an issue at all AFAIK

Do you actually need a tourist visa?

Yes, that is doable. You will want at least three hours--you have to build in a lot of time because your first flight could be late, plus you need time to collect baggage if you check it, wait in lines, etc.
Have you checked Turkish Airlines as an alternative? I would at least check multicity flights into IST and out of Rome, as it may be easier.
I purchased my Turkish visa on arrival in Istanbul, but I am not sure if anything has changed about the policy.

ETA Looks like they may have stopped requiring visa for Americans staying less than 90 days:
https://tr.usembassy.gov/u-s-citizen-services/u-s-citizen-services-faqs/
https://www.mfa.gov.tr/visa-information-for-foreigners.en.mfa

@ Christine

Yes Turkey has implemented an online Visa requirement for US Citizens. Its a simple form, fairly inexpensive and likely just a revenue generator.

@Valadelphia

Thanks for the answer. Its a long story, but due to limitations with using AA points for travel to Istanbul, we can save about $1500 dollars. We are planning to attend my son in laws family event in Istanbul but will be adding about 5-6 days in Italy and I can get a great deal by using AA points for the RT to Rome.

So my wife and I are debating on convenience vs savings. The other option is a Non Stop to Istanbul on Turkish but that's a $2200 one way for the 2 of us.

Perrmont-
According to the 2 links above-regular passport holders are exempt from visa requirement up to 90 days

Will you be checking luggage? If so, I believe you would have to pass through immigration, collect your bags, and then go through security again for the flight to Turkey, unless there is a transfer desk where you can re-check your bag.

Turkey is not a Schengen territory, so in theory you would just be a transiting passenger - if no checked baggage - and not have to leave the gate area.