Learn Hardcore Turkish: Bisiklet yolu üst geçidin altından geçiyor.

üst geçit + -(n)in → üst geçidin: genitive case, meaning of the overpass.
alt + -(s)I + -(n)DAn → altından: altı = its underside/bottom (3rd person possessive), then buffer -n-, then ablative -dan = from under. So the pattern is: X’in altından = from under X.

Why is it altından (ablative: from under) and not altında (locative: under)?

With geçmek (to pass), Turkish marks the path you traverse with the ablative (-DAn). You’re not statically located under the overpass; you are passing through that under-space.

Static: Bisiklet yolu üst geçidin altında. = The bike path is under the overpass.
Motion/path: … altından geçiyor. = … passes under (lit. from under).

Why does geçit become geçidin here?

Consonant softening. Many words ending in p/ç/t/k soften to b/c/d/ğ when a vowel-initial suffix is added. Thus:

geçit + -in → geçidin (t → d) Other examples: kitap → kitabı, ağaç → ağacın, kanat → kanadı.

What is the extra -n- in altından?

It’s a buffer consonant used when a case ending follows a 3rd-person possessive. Structure: alt + ı (its) + n + dan (from) → altından.
Parallel patterns:

evin önünden = from in front of the house (ön + ü + n + den)
arabanın arkasından = from behind the car (arka + sı + n + dan)

What tense/aspect is geçiyor, and could I use geçer instead?

geçiyor is present continuous (-iyor). In descriptions of layouts/routes, Turkish commonly uses this for what English states with a simple present: the bike path passes…
geçer (aorist) also works for timeless facts. It sounds a bit more neutral/general.
Both are natural; geçiyor feels a touch more vivid/in-the-scene.

Why is it bisiklet yolu and not bisiklet yol?

This is an indefinite compound noun (belirtisiz isim tamlaması). The head noun takes the 3rd-person possessive:

bisiklet (bicycle) + yol + u → bisiklet yolu (bicycle path) Other examples: yaya yolu (footpath), şehir merkezi (city center), öğrenci yurdu (student dorm).

How do I know whether bisiklet yolu means “the bike path” or “a bike path,” since there’s no article?

Turkish has no articles. Subjects without case markers can be interpreted as either definite or indefinite from context. Definiteness is overtly marked mainly on objects:

Bisiklet yolunu buldum. = I found the bike path (specific).
Bisiklet yolu yapıldı. = A/the bike path was built (context decides).

Can I write üst geçit as one word, üstgeçit?

Yes. Dictionaries and many style guides prefer üstgeçit (one word). You will also see it written separately (üst geçit) in real-life usage. With suffixes: üstgeçidin or üst geçidin.

Can I change the word order?

Yes; Turkish word order is flexible for emphasis:

Neutral: Bisiklet yolu üstgeçidin altından geçiyor.
Emphasizing location: Üstgeçidin altından bisiklet yolu geçiyor.
With a pause/comma for focus: Bisiklet yolu, üstgeçidin altından geçiyor. Keep the genitive phrase together: say üstgeçidin altından, not altından üstgeçidin.

Why use geçmek (to pass) instead of gitmek (to go)?

geçmek describes crossing/passing through or past something and governs the ablative: X’ten/altından/üzerinden geçmek.
gitmek is going to a destination and takes the dative: X’e gitmek. A path “passes under” something; it doesn’t “go to” under it.

What’s the difference among üst geçit, alt geçit, and köprü?

üst geçit/üstgeçit: an overpass (for vehicles or pedestrians). With pedestrian focus: yaya üst geçidi.
alt geçit: an underpass.
köprü: a bridge in general (over water/valleys/roads). Routes:
X’in üzerinden/üstünden geçmek = to pass over X
X’in altından geçmek = to pass under X

Could I say üst geçitten geçiyor instead?

That means “(it) passes through the overpass,” i.e., the overpass itself is what’s traversed (like a road using an overpass to cross above something else). Your sentence says the path passes under the overpass, so you need üst geçidin altından.

How would I say the opposite, that it passes over something?

Use X’in üzerinden/üstünden geçmek:

Bisiklet yolu nehrin üzerinden geçiyor. = The bike path passes over the river. For your noun: Üstgeçidin üzerinden would mean “over the overpass,” which is unusual semantically; typically roads pass over rivers/tracks via an overpass, not over the overpass itself.

Any pronunciation tips for these words?

ç like ch in church: geç-.
c like j in judge: in other words like geçici (not in this sentence).
ı (dotless i) is a back unrounded vowel, like the a in sofa: altından has two ı sounds.
i (dotted) is the front vowel: geçidin.
ü is like German ü/French u: üst, yolu.