I'm sure others will have more detail to give, but the answer seems pretty clear to me:
Unfortunately, the way to develop stronger breath support is with all the non-musical exercises your teachers have given you over the years. Here are a few:
Hiss like a snake. Sssssssssssssssssssssssssssssssssss. Until you run out of breath. You want to be able to give a constant sound output using constant muscular pressure.
That riff from Paul Simon's Graceland. "Somebody say: Ih - Hih-Ih - Hih-Ih (Somebody cry Why Why Why?)".
Long, slow notes. Similar to hissing like a snake, but a little closer to music. Same deal though: you want a steady tone at an even volume, provided by the slow contraction of the muscles of the torso.
It is important to be aware of the difference between diaphragmatic breathing and chest breathing. But be wary of overly-zealous advice to use only diaphragmatic breathing. All the muscles of the abdomen, chest, and even the shoulders, too, contribute to the shape and pressures in the lung. There's an exercise called the Full Yogi Breath, which works all of the breathing muscles together.
Projecting can also help. Put a picture of your girlfriend on the wall across the room, and sing so the picture can "hear" you.