At 30kph on an upright bicycle, 20W goes to rolling and drivetrain drag, and 180W to air drag. Now suppose we decrease vehicle drag so the same 200W will take you twice as fast. The rolling and drivetrain drag now takes twice the power, 40W, so only 160W is available for overcoming air drag. This scrubbing is the major source of friction drag. Thin tires generally have thinner tread and walls than fat tires, so the energy absorbed by flexing should be lower.
Narrow tires have less air drag.
I do not know how much drag is due to tire cross section, nor whether air drag is proportional to tire width. However, racing cyclists may be increasing rolling drag in order to eliminate even more air drag. One of the topics we discussed was the size of the 3 components of resistance. I have a way of determining these numbers for more general cases, and I will try to post something on that in the future. Note also that neither rolling or mechanical friction increase directly with speed.