If the small bridge is connected, the resistance of the circuit immediately decreases to a very low value. The current, thus, immediately increases heating the wire (a) which burns out.
This is called "short circuit".
In a lighting plant a short circuit can start a fire. This was prevented by using a safety device called "fuse".
In a fuse is a bit of wire or metal tape that overheads and melts (blow out) if the circuit is overloaded. The circuit is than open and the current can't flow.

This animation shows what happens in a short circuit.