Italian here, the exact same experience growing up. It was expected that you had a conversation this way. Why waste time waiting for each other to finish?
Unfortunately I developed a kind of intense mix of the two cultures and I have to work hard not to dominate others in (non Italian) conversations. I try to de-interrupt by reminding someone who stopped taking of what they were talking about and asking them to continue. This also isn't ideal because it's forceful but it helps.
In my experience, interrupters frequently assume something entirely different from what was actually being said, or even cut off the speaker before a "but" and wind up responding to the exact opposite of the intended point. This is needlessly annoying and confusing to almost everyone involved.