A verb I found while using Wiktionary that could (potentially) be used here is sillonner, though not being a native French speaker I don’t know how formal/informal this word is nor if it applies in this context.

For a line that keeps going in the same general direction, but wobbles back and forth, the verb that comes to mind is sinuer.

La voie étroite sinue le long de la rue.

Another possibility is serpenter, which means to move with an undulating motion like a snake. Serpenter is close to “slither” in English, but only refers to the shape of the movement, it doesn’t have any other snake-like connotation. Strictly speaking I guess the unmoving track sinue and the moving train serpente but in practice the two are largely interchangeable.

La voie étroite serpente le long de la rue.

Or you can just say that it keeps crossing from one side of the street to the other.

La voie étroite longe la rue en la traversant en de nombreux endroits.

La voie étroite passe sans cesse d’un côté de la rue à l’autre.

This is a complex situation, that is not being rendered precisely by a single term, and there is no fixed expression for that. "Serpenter" and "sinuer" are proper terms for the description of curved lines in general, but in this particular case saying that the track "serpente" in the street sounds vague: a street is something which, specially in old urban agglomerations, is not so rarely found to snake through the town or through a neighbourhood; so possibly one might ask oneself whether what was meant could not be that the track snaked through the neighbourhood along with the street. A clear rendering of the proper term—if there is a proper term, as user James K noticed "crisscross" does not appear to be a good choice—should involve a little description. Something as what follows would leave little doubt in the reader’s mind.

• La voie ferrée serpente dans la rue, la traversant obliquement pour passer tour à tour d’un côté à l’autre, ne longeant jamais ni l’un ni l’autre que sur une courte distance.

"que sur une courte distance" can be eliminated, replaced or modified to fit exactly the particular case; other modifications can be made (for instance the negative "ne … jamais" can be removed so as to have an affirmative form, etc.)

If after a crossing from one side to the other the track keeps running along this given side over a distance that is long enough, then I think that describing it as a meandering track is not very meaningful.

• la voie ferrée longe tantôt un côté de la rue tantôt l’autre, qu’elle atteint simplement en la traversant (au niveau de la chaussée).

"au niveau de la chaussée" can be added at the end to make clear there are no bridges or tunnels.

Knowing this type of rall line prior to this question (for example : the tram n°28 in Lisbon), the first idea that come to my mind was :

• serpenter. "Former, décrire une ligne sinueuse"

Le petit train serpente d’un bord à l’autre de la chaussée.

But, after thinking about it, some people would probably use the less formal and less accurate (curves are not zigzags)

• zigzaguer "Marcher, avancer en faisant des zigzags"

Le tramway zigzague d’un bord à l’autre de la chaussée.

Finally, reading the question again, it seems the verb crisscross here is describing the way the rails, due to their curve, are crossing the main flow of the street multiple times. The closest verb in that sense would be:

• s’entrecroiser

but the sentences with this verb seem less fluid :

La ligne s’entrecroise avec la circulation à de nombreuses reprises.

For this very example :

• La ligne de chemin de fer empruntait la rue, tantôt à droite tantôt à gauche.