Turns cause drag. Drag causes the nose to drop. If you don't pull back you'll descend.
Mmmmm. Turns don't cause as much drag as they cause the direction of lift to change such that there isn't enough lift in the vertical to maintain altitude.
Most of the additional drag comes from raising the angle of attack to maintain altitude in the turn. If you don't raise AoA, and descend during the turn, there's no significant drag created. Some. But not much. As long as it's coordinated.
The effect of loss of vertical lift causes us to pitch up to add it back, in order to hold altitude, and that adds drag.
That pitch input adds to both vertical and horizontal lift and has the effect of maintaining altitude while tightening the turn. Trading sideways lift for speed then becomes a function of bank angle chosen.
Roll it 90 and hold it there and push to keep the wing from lifting in the horizontal plane, and much of your lift is now coming from the side of the fuselage the side of the vertical stabilizer, and propwash from raw horsepower.
Roll it 90 and pull hard, and you can still stall the wing, and it's providing nearly no vertical lift when you do it.
There are also some aircraft that the nose doesn't drop much, if at all, with airspeed change. They'll bite you square in the butt if you don't pitch them where they need to be. They tend toward the aerobatic.
Put a Cessna in a hard slip, and the nose will fall and it will tend to want to maintain airspeed. Do the same in a Citabria and the pitch will tend to stay right where it was, and the aircraft will slow more, without a purposeful nose down pitch input.
The statement "turns create drag" is true, if the turn is descending less than the loss of vertical lift. Descend enough to maintain airspeed in the turn, and no additional drag really is caused by the turn.
This is why the power loss PUSH after takeoff if one thinks (or better, knows) they can turn back to the runway. It's also why glider pilots are taught that hard push on a rope break from day one. A glider pilot friend and I have been discussing this recently.
He asserts that power pilots would benefit greatly from a little rope break training in gliders to avoid the tendency to not PUSH IMMEDIATELY on power loss at takeoff. I think he has a very good point. He says any glider pilot who transitions to power will naturally do this PUSH if the engine quits on departure as their first reaction, and it's the right one. THEN you look for your landing spot.
If you have the altitude, and you KNOW you have it to make that turn, you must PUSH the nose down even more in the turn to trade that altitude for airspeed during the turn. If you started at best glide and turned, you MUST nose down to maintain best glide in the turn.
Glider pilots do it all the time. Commercial glider pilots do it even lower than Private glider pilots. They know if they have a 20:1 or even a 40:1 ship, how high they have to be to turn back. Many power pilots at 7:1 or worse, don't KNOW and haven't tried it, even at altitude. But you can still teach them to PUSH at power loss before even thinking about PUSHing even more to make that turn-back.