Why does one switch only work when another is on?

Remember that the power comes in through the black wire into the circuit and the neutral returns the power to the source. The switch is just there to interrupt the circuit or complete it. In the illustration, the switch on the left does not complete the circuit; creating a neutral fault.

FIX:

Step1: Looking at the diagram, connect all the white wires together using a wire-nut.

Step 2: Using a wire-nut and two separate black wires, split your (Black) power wire to connect to each switch (on only one black wire, to one screw, per switch). The black wire that comes from each lamp's circuit will have to be connected to either remaining screws (one wire, to one screw, per switch).

If your original wiring schematic is correct black and white on the switch this is a switch loop, or switch leg!

the power goes to the fixture. Then the hot comes back to the switch on the white it is not a neutral.

at this point do not connect all whites together as someone said. The white “always hot” should be marked as a hot the black complete’s the circuit to the outside light. The switch turns on and off this connection!

New wiring would need to be run the new light. You cannot tap a switch leg to provide power for another device!

Be careful accepting assistance on the internet because there are a lot of people that don’t have a clue I hope you have not damaged things wiring etc. This is a prime example of internet only electricians in a circle !