Spoiler: My answer
Let's call the lines l1, ... , ln, and give an orientation to each line.
At any point in time, can define Turbo the Snail's current segment with a n-tuple: if li is the line she's currently on, its ith value is 1 (resp. -1) if she's moving in the line's positive (resp. negative) direction; otherwise, it's 1 (resp. -1) if she's on the left (resp. right) side of li from the point of view of someone traveling along it in the positive direction.
Whenever Turbo turns left from li to lj, no value in the n-tuple changes - she doesn't cross, enter or leave any line other than li and lj, and the n-tuple's definition ensures that the ith and jth values don't change. Similarly, when she turns right from li to lj, the ith and jth are the only values that are changed. As a consequence of this, the product of the n values of the n-tuple is constant throughout Turbo's entire journey.
However, between a segment on li being passed both ways, the only variation in the n-tuple is the ith value being flipped, which means that those two types of movement have opposite values for the product. This means it's impossible to access them both in the same travel.
Turbo the Snail never visits the same segment both ways.
At any point in time, can define Turbo the Snail's current segment with a n-tuple: if li is the line she's currently on, its ith value is 1 (resp. -1) if she's moving in the line's positive (resp. negative) direction; otherwise, it's 1 (resp. -1) if she's on the left (resp. right) side of li from the point of view of someone traveling along it in the positive direction.
Whenever Turbo turns left from li to lj, no value in the n-tuple changes - she doesn't cross, enter or leave any line other than li and lj, and the n-tuple's definition ensures that the ith and jth values don't change. Similarly, when she turns right from li to lj, the ith and jth are the only values that are changed. As a consequence of this, the product of the n values of the n-tuple is constant throughout Turbo's entire journey.
However, between a segment on li being passed both ways, the only variation in the n-tuple is the ith value being flipped, which means that those two types of movement have opposite values for the product. This means it's impossible to access them both in the same travel.
Turbo the Snail never visits the same segment both ways.