As you can see from my picture there are a lot of places where the same lines run on top of each other. Imagine a tram map. How can I make sure that the ones that are behind are visible too?
Just like you see from metro or train maps:
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First you must use categorized symbols, as you did.
Then you can edit each line symbol (double click on it to open symbol selector, then click on "simple line" for more properties) by giving a specific offset. Note that the offset is defined based on the direction of digitization (positive offset = right side), so you may need to flip some lines in order to have a nice result.