Correct. You need to get "aquaticity", and this is better done in swimming pool, or even in your bath tub, simply using a snorkel. Submerse your face and breath through the snorkel. As the mammalian reflex is triggered by neurotransmitters close to your nostrils, you should start submerging just the mouth. Then the nose. Then the eyes. It can help to squeeze the nose with your fingers (which prevents water flowing in) but in the end you need to control your soft palate for locking the noise cavity and being able to breath freely from the mouth without closing your nostrils.Look up Mammalian Diving Reflex which might give you a better idea of what is happening. Some suggest practicing immersing you face in a sink full of water. I have suggested this to others with a similar problem and they found it helpful.
When I started diving, there were THREE MONTHS in the swimming pool before being given a rebreather (which, at the time, was a closed-circuit, pure oxygen rebreather).
We made months of exercises for gaining "aquaticity": and to learn free diving, which at the time was a prerequisite for scuba diving.
Now instruction is very different, it is assumed that people entering an OW course are already "fully aquatic".
This was not the case of the OP, evidently.
Although some instructors think that proper equipment can fix lack of "aquaticity", I am of of the old school, and I think that everyone can dive, but only if previously properly accustomed to the water environment. Starting straight with the scuba equipment is not the way to get this aquaticity.
You must start with no equipment (you must be able to stay face down and going down to the bottom of the pool with no equipment at all, and to swim something as 15 meters underwater with breast stroke), then introduce the snorkel. Then the mask (learning to swim and snorkeling with mask flooded, and later to evacuate it while snorkeling and while breath-holding), then the fins (learning to kick properly in different styles, but no bicycle, please), then the weights. then the suit. Then the tank, with no BCD. Then the BCD. Then the computer.
The idea of going through this in just a couple of days is nonsense for me...
My OW course lasted 6 MONTHS!
Later I worked as instructor in holiday resorts, and we managed to pack a proper training in just two weeks: the first week for aquaticity (no scuba equipment), the second for training to real scuba diving. But it was 2-3 hours in the pool in the first week, and 2 dives/day in the sea during the second week.