Basically, the problem is that what you have there isn't an OR gate. An OR gate needs to have some way to convert a weak input signal into a strong output signal, and a transistor all by itself can't do that.
If you want to build your own logic gates, you'll need to build them according to some logic family. One good option is resistor–transistor logic. The most straightforward way to build a real OR gate is to build a NOR gate and a NOT gate and connect them together.
I've designed this circuit that should work. You can simulate this circuit at falstad.com.
The red circle on the right is an LED, and as you can see in the picture, it's currently turned on. The switch at the very top is the power switch; if you turn that off and on again, you'll see that the LED stays off. The switch on the left side is the A input switch; if you turn that off and on again, you'll see the LED turn on again and stay on.
I had to play around with the resistor values a bit to get this circuit to work. The 470 and 220 ohm resistors that are connected to the power rail are output resistors; if those are too large, then the gate will produce a weak output, and if they're too small, then they'll get too hot. The 2.2 kohm resistors that are connected to the inputs are input resistors; if those are too large, then the transistors will fail to turn on and off correctly, and if they're too small, then they'll load the output too much. I don't know what resistor values would be best, but these values seem to work okay.