It depends on what you mean by the verb "turning."
No, you don't really need to steer your vehicle to take that turn. You don't even need wheels for that. When thrown at the correct speed and angle, your car will slide through the turn without any problem. As you said, banking will provide all the forces for the turn. But it won't be a pleasant experience for the driver and the passengers, though.
We do not bank the roads to steer the car; we do it to ensure that we are in a good position to do so.
When we say "turning," we also silently assume that the car will align its direction with its trajectory automatically. This is not something to be taken for granted, and many factors are essential for this process to happen in a controlled fashion. We need some way to change the orientation of our vehicle, as well as its direction of motion.
Banking road geometry
Imagine a car on top of an imaginary 2-dimensional rectangular sheet of plastic. As long as you don't stretch or compress the sheet, we don't change the geometry. Thus, we can call two surfaces geometrically similar if one can be formed from the other without stretching or compressing. The steering of the wheels depends on the road's geometry. Even if we bend the road as shown, the person driving the car won't notice a difference as long as we do not change the local geometry. (Read more about intrinsic and extrinsic curvature)
You can now see that a 90-degree banked road is the same as a straight road for the car geometrically. You don't need to turn the car here.
Ideally, if all our roads were constructed like this, there wouldn't be any need for turning. But given our primitive public transport system and city planning, it is what it is for now. Coming back to reality, a 45-degree banked road is not the same as a rectangular straight road.
You will need to stretch some part of the plastic sheet in order to form this. If you make this banked road flat, that geometrically similar surface will look like this.
Here, you will need to turn your wheels like you would on a normal turn. Now how would turning the wheels make the car change its orientation?
Steering is the process through which we change the direction of motion as well as the orientation of our vehicle. We make use of some properties of wheels to achieve this.
Wheels like to move precisely in the direction they are pointed towards. (This is a bit more complicated because real tires are tori, and the direction would depend on the point of contact.)
Now if you have a bike, we have two wheels trying to move like that. When you are moving straight, there is no problem because both wheels are moving on the same path. But once you turn the front wheel right, you have a situation where the front wheel wants to go right, but the back wheel wants to go straight. The only way this can be achieved is though the change of orientation of the bike. The bike moves in a circle so that both wheels are tangential to the circle.
This is how steering works. In addition to changing the direction of motion, notice how the vehicle's orientation is also changed.
Now consider a pair of this arrangement, as in a four-wheeler. For the same turning angle, you will immediately see that the left and right wheel pairs try to move in a separate circle. In order for this to happen, the chassis needs to break, or the wheels need to slip.
The only way to solve this problem is to make both wheels turn differently. If you are wondering, yes, the front wheels of your car are not parallel during turning. During a turn, the outer wheel turns less.
This is a well-known problem in mechanical engineering and was solved by a German carriage builder called Georg Lankensperger in 1816. You can read more about this here, for a start.
Image credits: Wikipedia
TL;DR: Your car is always moving along the arc of a circle formed by the point of contact of its four wheels.
If wheels do all the turning, why bank the road?
When you take a turn, your car is moving in a non-inertial frame of reference. The centrifugal force associated with this motion tries to throw the car away from the road. Usually, on flat roads, the friction between the tires and the road counteracts this centrifugal force. This friction has a limit to it, decided by the weight of the car and the coefficient of friction between tires and the road. But the centrifugal force can be arbitrarily high, depending on the curve's radius and the vehicle's speed. So there is a possibility that the centrifugal force overcomes a friction threshold, and your car flings out of the curve. This is why we have speed limits on curves.
Banking the road provides an another force in addition to the friction to counteract the centrifugal force: the normal force. On flat roads, normal forces are acting perpendicular to the centrifugal force, making it effectively useless. But on a banked road, the normal force is tilted, giving us a helpful component against the centrifugal force.
This increases the threshold for centrifugal force to overcome. Hence we have larger speed limits on banked roads. Also, as pointed out by @Lawnmower Man in the comments, banked roads are much more comfortable for the passengers in the car. Humans inside the car are like inverted pendulums with those heavy heads. Banking helps reduce the torque acting on them, lowering their vulnerability to tip over. Same reason it is wise to crouch your heads on a merry-go-round (unlike these idiots below).
Banking the road ensures that we are in control. Banking the road ensures that all our geometry works properly.
I hope that this now answers your question. Have a nice day!