It actually kills headlights (when starter operated) regardless of whether in dip (low) or main (high) beam
I would suggest it would be unlikely to be the headlight-cut/starter relay - that uses the normally closed contact so doesn't require to be energized to complete the circuit - the contacts would have to be severely pitted or oxidized to not make the circuit. But not impossible either.
If the starter runs, then nothing wrong with fuse or the key-switch as those are common for both headlights and starter circuits;
Have you owned this bike since new? One common mod is a headlight relay kit - if one of those was fitted by a previous owner, they have their own supply fuse and that may well be blown
Otherwise as TURBO200R4 suggests, it could be problem in the switch