The switch in your bathroom has not a "neutral wire". The white and black wires you see in the switch are both playing the"hot source".
From the light, the hot source is run using the "white wire to the switch, and from the switch, the "black wire" will bring "hot" power back to the light.
There is no "black/black/bare" cable in the market, and for this reason the white wire has been used to run the "hot" source power.
Now, in order for you to run power to the additional light.
Turn the power OFF.
TWO MAIN OPTIONS
1) You must run from the existing light box a cable to the new light location, and install a box.
From this box you will do exactly the same as you see in the existing connection. You run a cable to the switch box.
2)- Connections from the original box to the new box.
In the original light box, you will see one or more black wires together, to which a white wire is also attached. This is the hot source, so you connect the black wire to this source.
Connect the new white wire to the white wire(s) of the original light box.
Connect the bare wire to the other bare wires.
Run a cable from the new light box to the switch.
3)- Connections from the new box to the switch.
Connect the white wire going to the switch to the black wire of the new light box.
Leave the white white alone.
Connect the bare wire to the another bare wire.
4)- Connections on the switch box.
Install another switch.
Here you have also two more options. You can use a single box double switch, or replace the single box for a double electric switch box.
In this case, I would prefer to replace the single switch box by a double switch box.
The installation to the switch is the white wire to one of the terminals, and the black wire to the another terminal. The bare wire to the green screw on the top of the device. Off letters at top On letters at bottom when installing it to the box.
a)- Do not use three or four ways switches. The switch must be single pole and must show the ON and Off position
b)- Do not try to be "sneaky" and run just a single wire from the existing switch box to the new light to provide power. This method is "illegal", because you are supposed to run a cable not so a wire inside the walls when is AC power.
On the new light box, you connect the incoming white wire to the white wire of the fixture, the incoming black wire to the black wire of the fixture, and bare wires together.
There is also another method to run the cables legally.
1)- Replace the existing (black and white) cable by a three wire (black, white and red) cable from the existing light box to the switch.
Run a two wire cable from the existing light box to the new light box
1)- Connections from the existing light box to the switch box.
Replace the single pole switch for a single double switch.
On the light box, install the outgoing black wire to the other(s) black light wire(s).
On the double switch, install the black wire to the incoming"common" screw.
From the double switch, install the red wire to one of the outgoing screws, and the white wire to the another outgoing one. Use black tape and mark the white wire as a wrist band, so you know that this white wire is a "hot wire".
2)- On the original light box.
Install the "incoming from the switch" red and white wires as it follows.
Red wire to the black wire going to the new light box.
Put the black "wrist band" electric tape to the incoming white wire. (the wrist band is installed about one inch from the end of the wire. Connect this wire to the black wire of the existing fixture.
Connect the white wire(s) to the existing fixture.
Connect the white wire of the new light box to the group of white wires. (all white together but the one with the wrist band)
Use wire nuts, and be sure that there are no loose connections.
3)-On the new light box, the installation to the new fixture is black with black, white with white, bare wire with bare white.
Return back the power.
Test the work.
If everything is OK, then finish the installation of the fixtures to the wall and the switch to its box.
If the construction is new (with drywall panels), running cables won't be too hard. You might need to make holes to the studs inside the walls.
There are some Youtube videos showing how to install new electric boxes in walls. They might be a great help for you to have a better idea of how to make the job with lesser destruction.
Besides electric supplies, you will need plaster and plaster paper to patch up the holes. When you make the holes, make them no bigger than 5" by 5", enough to introduce your hand and fish the cables. The same squared cut piece can be used to path back the wall.
You just need wood 1"x1 1/2" to be used as a"stud" . You cut the piece of wood at least 9 inches long. Put it inside the hole and pull it back against you. Using drywall screws, install a screw one inch out the edge of the hole to the wood in both opposite sides. Now you have the wood attached to the wall. You put the 5"x5" cut piece of drywall over the "stud" and use a drywall screw to attach it. After that, pour plaster on the edges about 2" around, and install the paper. The techniques about plaster are also shown in Youtube videos.
If my instructions are confused, bored, or good for nothing, then don't worry about it, this message probably wasn't for you but for the another reader.