You're trying to use the AND gate to source current into the base of one of your OR gate's inputs, and it's going to be really hard to control the amount of current that flows when the AND gate is off, because the input impedance of the OR gate will affect it.
It's easier if you think of each gate as an individual building block that connects directly to the two rails and produces an output from them. This allows you to manage the flow of current more easily, because you can control the impedance of each output to ground, and thus create a preferential path for the current to follow when a gate is shut off. For example,
this circuit implements two AND gates feeding into an OR gate, whose output is then inverted by a NOT gate. As you can see, there is still stray current in the circuit (that must be the reason why all those TTL chips tend to get so hot
), but it's easier to control because each gate is independent from the others.
I'm sure more experienced folks on the forums will be able to provide better solutions/explanations, but I hope this helps a bit. Cheers!
—CC