If it is not blowing fuses....it is not a short. A short is when a hot wire comes in contact with a ground, and it blows a fuse.
You either have a broken wire - or a faulty switch.
You need to go to the Technical Section or your Repair Manual and look at the Wiring Diagram.
http://suzukisavage.com/cgi-bin/YaBB.pl?num=1294779818The power is taken up to the left handlebar with an orange wire.....that take the power up to the left handlebar control and the control sends the power to the horn, turn signals or lights. If the headlight goes out - but the horn and turn signals continue to work, then you don't have any issue with the orange wire.
You then need to check the switch. Take the handlebar control apart and pull it off the bar. You might first test to see if the wire into the headlight switch has power to it (I believe the wire is black with red stripes). If that has power to it, then the yellow wire coming out of the switch is the high beam and the white wire is the low beam - and you need to test those for power. The white one will have power when the switch is on the low position, and the yellow wire will have power when in the HI position. If you have power to the red/black wire - but can't get any to the yellow or white....try some Tuner or Electrical Contact cleaner. If you can't get power to the yellow or white wires - but the red/black wire has power...you need a new switch.
If the power goes to the yellow and/or white wires - but the headlight still doesn't work, you need to test the yellow and white wires at the headlight. If you have power to those but the headlight still doesn't work - then it is either bulb or the black ground wire is broken somewhere. If you have power at the yellow and white wires at the handlebar - but not at the headlight....then both of the wires are broken somewhere in the harness between the left handlebar and the headlight.
Dave