eracode:
Gordy7:
Remove the bracket.
Grind off the protuding screw shanks flush with wall.
Reposition bracket and pre-drill a new set of holes.
Lube the holes with CRC or similar. (soap applied to screws was the old way)
Get some decent screws.
Don't over torque the screws... In fact use a manual screwdriver to keep a feel for screw insertion.
Edit: Reads like the OP was fitting 2 brackets... in which case relocate both brackets.
This would be a good way to go - but only if you can’t get the broken screws out reasonably quickly and easily. A lot more work in the grinding approach and it may not be necessary.
Looks like the broken screws may have seized in something really tough in the wall... so removing the broken shanks may be really difficult.