The cap will decay to 37% of its initial voltage in one time-constant, which is R x C. If the lamps took 1A, giving a resistance of 83Ω, and assuming their resistance doesn't change with voltage (although is will reduce some as the voltage drops), then for a 10 second time constant you would need a capacitance of 10/83 = .12 farad or 120,000µF (26 of the 4700µF capacitors you used in parallel).
And of course, your lamps probably take more than 1A so the required capacitance would go up accordingly.
So just connecting a capacitor to get your desired delay is theoretically possible, but rather impractical with available capacitors. You could conceivably use a super capacitor, since they come is farad sizes, but they are usually low voltage devices (typically about 2.5V), so you would have to put about six in series to get the 15V rating required for your car's maximum battery charging voltage.