Towards north or towards the north

Question

Let's say there is a region with two parts, north and south. This region has a railway and has been urbanized, which can now travel towards northern train line, crossing over a river.

The railway from the southern part of XYZ region has just been extended, which can cross over a river towards the northern train line.

Or

The railway from the southern part of XYZ region has just been extended, which can cross over a river towards northern train line. (please see the picture on the right)

(Added sentence after the comments and an answer)

*The railway from the southern part of XYZ region has just been extended, which crosses over a river towards north.

Answer 1

In your example, since you are designating a specific reference you should use "the"

cross over a river towards the northern train line

Expressed in this way, it may mean that there is different train line which lies further north.
To say it is the same train line extending north, you might say

cross over a river on the train line north

Since you are referring to the railway (route) and not just the train itself, it might be more natural to say

which crosses over a river

since the railway actually does cross the river, stating ability ("can") is awkward.

To express action and direction, a native might simply say

crosses over a river going north.

[ADDITIONAL EDIT]

They went north.
They went going north.
They went towards the north.
They went northward.

Thw way your new sentence is constructed, you are saying the railway crosses over the northern side of the river. "Towards" only describes the direction faced, you need a verb to show movement.

which crosses over a river and travels north.
which crosses over a river and goes north.
which crosses over a river and goes towards the north.