Motor transport & units in ww2
I don't think there were real mechanized divisions in the WW2. Even the most advanced APCs were of limited mobility cross country and were only able to provide a very feeble fire support.
In fact the motorization question was very debated in the 30's and many different unit compositions were tried mixing tanks, APCs and or trucks, cavalry, motorcycles, motorized artillery, infantry and cyclists.
They tried to solve two problems: one was that the tanks alone could be devastating in the attack, but useless in cities, woods, etc, and besides couldn't hold the terrain once the attack was over; once and again tests and actual combat experiences (in Spain, for example) proved that tanks alone were always forced to retreat when they were short of fuel or ammo. The second problem was that tanks were a new weapon, and older ones like cavalry naturally didn't like the newcomers, and had plenty of influence to fight against them; some way had to be found to make motorization 'socially acceptable'. These divisions, usually baptized as light or fast, were one solution.
You can think that they were improvised, poor solutions or even somewhat funny, but even in 1945 the soviets were using units that mixed cavalry, armour and motorized infantry in this way with success. And bycicles were used with success so many times that one wonders why their use wasn't more widespread.
The italians had motorizable divisions, which were trained and prepared to use motor transport, but didn't have it permanently included. The idea has sometimes been laughed at, but U.S. infantry divisions were very much like that, too.