I agree with Limax2. I think you've got voltage drop from unwanted resistance somewhere in the circuit. You have to do your voltage check when there is a load on the circuit. That means the circuit must be complete and there must be current flowing through it for your voltage measurement to be valid. So make sure the bulb is plugged into the connector, turn the ignition on, turn the high beam switch to on, connect the positive of the multimeter to the positive high beam connector, connect the negative multimeter to the negative side of the bulb. On a good circuit you should see 12V, on yours you will probably see a lot less than 12V. That shows there is an unwanted voltage drop somewhere in the circuit caused by unwanted resistance.
To find out if the voltage drop is on the positive side of the circuit, keep the positive lead of the multimeter on the positive side of the bulb and connect the negative of the multimeter to the positive of the battery. If the positive side of the circuit is good you should see near 0V. If you see a couple of volts, there's a problem on the positive side of the circuit. Next do the same test but on the negative side. Keep one lead of your multimeter on the negative side of the bulb and the other lead on the negative of the battery, if you see a couple of volts there's a problem is on the negative side.
The problem could be a chaffed wire, corroded wire, corroded connector, burnt wire or burnt connector something like that. If both high beam and low beam are effected the problem must be on a wire that is common to both circuit so that narrows it down. If you do the voltage drop test on both sides of the circuit that'll narrow it down more.