Actually, the water vapour in car exhaust is mostly generated in the combustion process as far as I know. Even if there´s absolutely no water contained in the fuel or in the ingested air, vapour will be formed from oxygen in the air and hydrogen that was chemically bound in the fuel (which contains numerous hydrocarbon compounds).
A significant part of an oil combustion engine´s power is actually generated by burning the separated hydrogen to water, while the carbon will burn to carbon oxides (mostly carbon dioxide, hopefully).
I don´t know too much about the exact mechanisms in forming jet contrails, but my guess is that it´s a combination of a relatively high "base humidity" in the ambient air in conjunction with the additionally generated vapour from the combustion process which pushes total humidity beyond the dew point (it would have had to be close enough to it).
Since the turbine exhaust is hot initially, the actual condensation only starts when the exhaust has cooled off sufficiently. View Large View Medium
Photo © Josef P. Willems