When an aircraft is in level flight, there must always be a component of lift equal to the weight of the aircraft that's pointed vertically away from the ground. If I roll to 45º angle of bank, I'm can resolve the lift vector into two components that point away from the ground, and into the direction of the bank. At 45º they're equal, both at 68% of the lift component. But, as I've said that I need make a vertical component equal to the weight to stay level, that means that I'm no longer producing enough overall lift. To increase the lift component sufficiently, I'll actually need to make 1.47 times as much lift (i.e. pulling 'g'), which will leave equal components of 1g, both vertically and into the bank. The 'g' that's our component into the bank is what makes the aircraft turn. Within the aircraft though, we'll simply feel a bit heavier, as the 'g' we experience remains pointed at the floor.
G required for a level turn vs bank angle:
20º 1.06g
30º 1.15g
45º 1.47g
60º 2.00g
70º 2.92g
80º 5.76g
84º 9.56g
The "g" is equal to 1/cos(bank angle).
Looking at the way the g builds up, you can see why turns don't exceed 30º angle of bank.
So, could we turn without banking? Sort of. We'd still have to generate a force into the direction that we want to go. The only way that could be done would be to use rudder to point the aircraft. That would generate a force ACROSS the cabin. Everyone is used to having things fall to the floor, but having them go sideways (including yourself) would be disconcerting, uncomfortable, and dangerous. Added to that the fact that we couldn't generate sufficient force to have the turn happen any time soon, or that the tail would quickly over-stress and fall off.
The rudder isn't used to turn the aircraft, but rather to keep it pointed into the airflow. They're large so that we can generate the force needed when an engine fails. The only time you should experience a rudder only turn, is during a crosswind landing, when rudder is used in the last couple of seconds to point the aircraft out of the airflow but down the runway.