The fact that you have multiple passports does not change the fact that you are the same person, so the answer will be the same as if you did a visa run and returned with the same passport.
And the rule is clear, on the visa-free regime, you are allowed 90 days per visit, 90 days within a 180-day period. So after staying 90 days, you have to stay out of Turkey for 90 days before returning, or apply for a different type of visa that would allow you to stay longer.
Note that the rule says "90 days", not "3 months". So for instance if you arrived on 25 August, you are allowed to stay until 22 November, not 24 November.
Note also that if you stayed in Turkey at any time in the 90 days before the start of your current stay, you may already be overstaying.
PS: the Turkish government page here lists the 90 in 180 days maximum for many countries, but not for Italy (and a few other EU countries, but definitely not all). Not sure if it's an error or if there is actually a difference (there may be some bilateral agreement that supersedes the usual rule).
On the other hand, this page says:
The length of stay provided by visa or visa exemption cannot exceed 90 days within each 180 days. The regulation of 90 days of stay within the last 180 days is binding for all foreigners that will travel to Türkiye
Nonetheless, even when there is no explicit rules about it, there are usually less explicit rules like "you are supposed to be a temporary visitor and not attempt to live in the country for extended periods of time through repeated or successive visits", so instead of having a clear "you're not allowed to do that" you may end up in a much muddier "it depends on the bordel official".
Visa runs used to be quite the norm for many people in many countries, but in most countries this has been severely curtailed.