How can I extend an electric wire without a large cross-section connector?
Use a crimp joint with heatshrink
Join the wires with a crimp. This is a permanent means of connecting wires that, done properly, will never come loose. It involves crushing another piece of metal into the wire with a special tool.
It is, however, critical that you use the correct size crimp for your cable, and the correct tool. Test the crimp by trying to pull it apart - they should hold at least 5-10kg force.
Insulate with heat shrink tubing. This is a special type of tubing that can be slid over the cable then heated, causing it to shrink radially and seal onto the wires/cables. Ideally you would use one sleeve over each individual crimp/conductor, then another larger diameter layer over the sheath of the whole cable (assuming you are joining sheathed cable).
Apply heat using a hot air gun or blow torch - avoid applying too much heat.
The shrinking effect prevents the sleeve from moving, and reduces any gaps. Glue lined heat shrink is available if the joint needs to be watertight.

Image courtesy Smial of Wikipedia under CC BY-SA 2.0 DE.