I don't really understand your question. You said "So if A goes through it it will come as A' but what happens when it gets into line with the other A'?. So far, this is the only question you asked.
The answer to this question is that in the diagram you are looking at, there are only "connections" at the end of lines. So where A crosses over A', there is no connection, no interaction.
The output function of the diagram you posted is:
Out = A'BC + AB'C + ABC'
IE the output is true if any only if only one of the inputs is false.
So that was part 1 of what you were asking.
To represent this as a Turing machine, you need to decide what "states" you have. It seems clear that what matters is "which input have I seen before being false?". This means that you need states that tell you which of the inputs have been seen being false. The possible combinations are:
- I've seen no false inputs yet
- I've seen A' only
- I've seen B' only
- I've seen C' only
From each of these states, you will get to one of the others, or the final answer.
The rule for state 1 will be "if the input is false, then move to the appropriate state (2,3, or 4) and move the tape right.
The rule for each of the other 3 states will be "if the input is false, and it's not the one I've already seen, then reject. If the input is not false, then just move the tape right.
If you need reach the "accept" state, or halt after you've processed three inputs, then you need a few more states so you can keep track of how many you've seen already.