Transformer and magnetic field/flux through the iron core?
I hope I can explain this in an intuitive way. Think of a single coil and how it's flux field would look. All of it's inside flux flows in one direction and must loop back upon itself.
Now insert something that can support the magnetic field like a transformer core:
The field is continuous so it must loop back on itself. The magnetization due to the current in the primary coil runs all the way around the ring.
The only way to make the flux flow like you guessed would be to loop the primary around the entire core instead of one section. The section of the core that has no coil around it could not also generate it's own flux in the same direction because it would be adding to the overall flux field without a source of energy.
EDIT: Another thing that might be confusing you is that in a standard N/S magnet the driving energy is molecular. The small magnetic fields around each molecule have a net flux field. In a magnet, these molecules line up adding a little bit of flux to the other molecules. As all these molecules line up in the same direction all their flux adds to produce a field in the same direction throughout the metal. This is a different effect from an externally generated flux field from a coil as you can see from the pictures above.
EDIT 2: Think of a thousand tiny N/S magnets inside your material. I made a little drawing to show the N/S and how it would look. Sorry for the quality, I'm not a graphics artist and I couldn't do the corners where the N/S bends occur.
At first they are random and at rest with no net magnetic field. Then you apply a field using your coiled wire. Where you've wrapped the coil around the metal they align with the strong field. This in turn rotates the others to try to line up N/S throughout the metal (with some bending around the corners...not shown well). In your second drawing you require the fields to stay at 90 degrees in the top piece, 180 degrees in the right piece then 270 degrees in the bottom piece in opposition to the internal fields.
This is how I like to visualise it. If you imagine you have a core like in my first image then the flux lines are like what you would expect. Now remove the right half of the core and your're left with your transformer image. There is still magnetic flux in the air, but it is left out normally as it is negligible in comparison with the flux in the core.
Hope that helped.