Is this normal?
No, but it is not especially unusual.
Do I have a problem with my wiring setup?
Insufficient information to tell.
Do LEDs behave this way?
It is a common problem with some types of LED "bulb" in certain situations.
LED replacements for incandescent lights are relatively new so a lot of house wiring and accessories that people have already installed are designed to work with incandescent bulbs, not with LED replacement bulbs.
Some older accessories that need to draw a small amount of power at, for example, a switch location that has no neutral (only live and switched live) do so by drawing a tiny current through the light bulbs controlled by the switch, the current is too small to make an incandescent light bulb glow but is enough to make most kinds of LED bulb glow.
The working part of Incandescent bulbs, (the coiled tungsten filament) operate at 120 V AC or 230V AC and draw around 500 mA current when working.
Each of several working parts of a LED bulb (each individual light emitting diode itself) operates at about 2V DC and draws maybe 8 mA from the 120 V side of it's driver circuit.
So it can only take a very small current to make a LED begin to light up.
Since LEDs are low-voltage constant-current DC devices, there has to be some driver circuitry somewhere to make them work with 120V AC. This circuitry is either built into the "bulb" or is a separate box. There are many different designs for the driver circuitry and some of these are more susceptible to producing the sort of effects you report.
I don't know if induced voltages from long runs of wiring next to each other are enough to start to drive a LED, but there are probably many circumstances where this happens unexpectedly - it isn't a certain indication of a wiring fault. Far from it.
I tested with a voltage sensor, and it does indeed beep for a split second when hold it up to the bedroom light wire while turning on the bathroom light.
This does suggest some sort of inductive transient voltage being generated. Are there any fluorescent lights in the bathroom?
To properly diagnose this issue you will probably need more equipment - or hire an electrician.
From the color of the wires I assume you are in either Australia, New Zealand or the UK if the home was built prior to 2004 which it certainly looks to be.
The red and black wires are your hot and neutral lines respectively. Since there is a big junction of three sets, I am guessing they are feeders and are always live. Your light is connected across them which is why it remains on. That being said, what is that blue wire doing by itself?
It looks like the lonely blue wire is paired with the brown wire. They look like they form whats called a switch loop. The brown wire is the feed to the switch from the red hot wires. The power is then fed back via the blue wire.
If you have a voltage tester, turn the switch on and test from the single blue wire to the black leads. If it shows power, turn the switch off and repeat the test. If you don't have power, you found the switch leg which is the blue wire. You then move the brown wire of the cable going to the lamp to that lonely blue wire on the terminal strip.
If you don't have a voltage tester, don't worry. Just move the brown lamp wire to that blue wire and turn your switch on and off. Even if its the wrong wire, nothing should blow up.
Edit: added a picture for clarity![Move wires](https://i.stack.imgur.com/bQ7Wk.jpg)
Best Answer
Most likely the bedroom light is wired downstream from the bathroom light, but the wiring in the bathroom switch box is wrong. To investigate further, turn off the circuit, remove the switch wallplates, unscrew the switch from the box, and take a good look at the wiring. Post a picture here for more specific help.
Most likely problem: the bathroom switch box has some mixed up wiring that can be fixed by connecting the wires that are already there in a different way. This would apply if that switch box has three cables (each a 3-wire bundle: black, white, ground) coming into it. One is the power supply, one goes to the bath light, and one continues to the bedroom. If that's the case then your problem is that the bedroom is wired to the switched power instead of to the constant supply power.