There was no TSA issues whatsoever. The issue was with AA plaine. I booked the flight knowing that I'll be traveling in Boeing 752 which is quite large plain with three seats on each side. On the entrance to the gate where you check for boarding pass, the crew had fitting slot and my luggage with packed turntable did NOT fit. I removed TT out from the luggage(was packed securely in carry-on luggage) and put into there my backpack carying turntable in my hands. When I stepped into the plane, I realized that the carry-on luggage space is substantially smaller than in those smallish Embryers or regional jets. It barely can fit the purse!
My flight back was in US-Airways A-319 which fit both: my carrier and my backpack with no issues.
I figure that AA pushes passengers to check their luggage larger than purse for extra $$. AA for me translates to a-a which means no go:-)