Did you just arbitrarily number those pins? If so you are lucky as that is their correct pin out.
OK the 1 & 3 have power, this is good, pin 2 is grounding through the bulbs this is normal (unlit bulb is a direct short) pin # 4 should be a ground, this is the pin that leads to the switch and it is not what it should be the relay is not suspect at this point as it is not getting all it requires to operate properly, this does not grantee it is good (likely is good) just that you have other problems preventing it from working weather it is good or bad it cannot work with support from other components
Possible causes:
A bad headlight switch
B bad diode
C bad transistor (light retainer relay)
D bad ground to switch
E bad wiring in-between A-D
Both the diode and “light retainer relay” are hard to get to so next you should get access to the switch and its connector for further troubleshooting,
To test the switch reinstall the head relay and ground pin 13 (black and orange see diagram for pin out) of the headlight switch connector the headlights should come on, if they do then everything up stream is OK, if not post up and we will move on,
If grounding pin 13 works next test the ground by shorting pin 13 to pin 11 (one of the two white & black wires) the headlights should come on, if they do the headlight switch is bad, if they come on when 13 is grounded to bat – but not when to pin 11 the ground for the switch is bad, if they never come on more troubleshooting in needed but you have the problem cornered to in-between the headlight switch and headlight relay there is nothing expensive between the two but it is all hard to get to
If you need headlights for now you could fashion a jumper with spade terminals to short pins 1 and 2 of the relay socket with the relay removed, this will turn on the lows, you will have to remove the jumper every time you park as they will not turn off with ignition, to make the jumper you will need a crimpier or soldering iron some fairly heavy wire (12 gauge or so?) and the matching male spade terminals