Removing bulb causes a different fixture to go out
It's difficult to tell for sure, as I'm not sure where all the wires in the box go. But it looks like the lights are run in series, rather than in parallel as they should be.

To fix it.


Turn off the power and verify that it's off.
Connect all the black wires together in one twist-on wire connector.
Connect all the white wire together in the other twist-on wire connector.


This image shows how electricity moves through the circuit.



Notice that the neutral wire "coming back from" light B connects to the hot wire "going to" light A. Then the neutral "coming back from" A, connects the the other circuit neutrals.

The only way that this wiring would make sense, is if B was a switch controlling A.

removing the bulb from either A or B interrupts the circuit, and causes both light to go out.