The current would encounter the resistance twice, and the double resistance would halve the current, so that the bulbs would glow only feebly. Connecting the bulbs in parallel means that electricity goes through each bulb only once.
A headlamp bulb, for example, is designed to have a degree of resistance so that it consumes a certain current to glow normally. But there are at least two headlamps in the circuit. If they were connected in series, electric current would have to go through one headlamp to get to the other.
If the cable overheated, there could be a fire. The fuse prevents that happening, because its thin wire will melt and break the circuit long before the cable itself can heat up and burn. Some cars have only two fuses. One rated at about 30-50 amps protects components wired through the ignition switch — flashers, wipers, heater-motor and instruments.
Where wires run side-by-side they are bound together in a bundle, in a plastic or fabric sheath, to keep them tidy and less difficult to fit. This bundle of wires stretches over the length of the car, with single wires or small groups of wires emerging where necessary, and is called the wiring loom.