Another explanation:
The total drag curve is more-or-less U-shaped, with Vy/Vbg at the bottom of the curve. On the left hand of the curve, most of the drag is caused by induced drag, at the right hand of the curve, most of the drag is caused by parasite drag.
Anytime you fly at a speed above Vy/Vbg, and you either increase drag (induced, by pulling g, or parasite, by for instance sticking your hand out the window), your speed reduces and will get closer to Vy/Vbg. Anytime you increase thrust the reverse happens. In this part of the curve the aircraft will automatically find speed stability due to the way the curve lies.
What the OP describes is actually the situation that you are flying below Vy/Vbg. Due to the increased induced drag, caused by the g-loading in the turn, the total drag increases with such an amount that the aircraft slows down even more, which requires even more alpha to compensate, which increases the drag even further. Up to the point where the aircraft stalls.
Try entering a steep turn with a speed that's less than Vy/Vbg, and then hold the angle of bank, power and altitude constant. I give it about 15 seconds before you stall. Make sure you have plenty height under the wings because you may end up in a spin or spiral dive.