Turing machines only model computation. Real life is interaction. Check the work of Peter Wegner. When interaction machines enter into the picture, AI can be embodied, situated and participate in adaptation processes. The emergent behaviour may bring AGI in a pragmatic perspective. But interaction is far more expressive than computation rendering theoretical analysis challenging.
Interaction is just another computation, and clearly we can interact with computers, and also simulate that interaction within the computer, so yes Turing machines can handle it. I'll check out Wegner.