Learn Hardcore Japanese: kodomo no kami ha hayaku nobimasu.

If you used が (子供の髪が早く伸びます), it would sound more like you are identifying what grows quickly (for example, in contrast to something else):
It’s children’s hair that grows quickly (not something else).

So:

は → topic, general/contrastive statement
が → subject, often used when introducing new information or focusing on who/what does something

Why is it 早く and not 早い?

早い is an i-adjective meaning fast / early.

When an i-adjective modifies a verb (to say “do X quickly” or “do X early”), it usually changes form:

早い → 早く

This -く form functions like an adverb:

早い髪 = fast hair (adjective modifying a noun; a bit odd conceptually)
早く伸びます = (hair) grows quickly (adverb modifying a verb)

So 早く伸びます literally means “(it) grows in a fast way.”

What is the nuance of 伸びます here? Does it mean “grow,” “extend,” or something else?

伸びます is the polite non-past form of 伸びる, an intransitive verb meaning to grow, to lengthen, to stretch (by itself).

In the context of 髪 (hair), 伸びます means “(hair) grows / gets longer.”

Some common uses of 伸びる:

髪が伸びる – hair grows / gets longer
背が伸びる – (someone’s) height increases / someone grows taller
影が伸びる – a shadow stretches

Don’t confuse it with 伸ばします (伸ばす), which is transitive and means “to grow (something) out / to let something grow / to extend something”:

髪を伸ばします – I grow my hair out / I let my hair grow

Why is it 伸びます and not 伸びています? In English we often say “is growing.”

Japanese uses the non-past (伸びる / 伸びます) to express:

habitual actions (something that regularly happens)
general truths / facts

Here, the sentence is making a general statement: Children’s hair grows quickly (as a rule).
For that, 伸びます is natural.

伸びています (progressive/continuous) would emphasize that right now the hair is in the middle of growing:

子供の髪が早く伸びています。= The child’s hair is growing quickly (right now / recently).

So:

伸びます → general fact / habit
伸びています → current ongoing situation

Does 子供 mean “child” or “children” here? There’s no plural -s.

Japanese usually does not mark plural with an ending like English -s.
子供 can mean:

a child
children (in general)

In 子供の髪は早く伸びます, the most natural interpretation is a general statement:

Children’s hair grows quickly.

If you wanted to emphasize many children, you could say 子供たち:

子供たちの髪は早く伸びます。= The children’s hair grows quickly.
But even without たち, the original sentence already works as a generic “children’s hair.”

How would I say “My child’s hair grows quickly”?

You can specify “my” with 私の, though in real conversation it’s often omitted if the context is clear.

More explicit:

私の子供の髪は早く伸びます。
My child’s hair grows quickly.

In casual spoken Japanese, people might just say:

子供の髪は早く伸びます。
If it’s obvious you’re talking about your own child, 私の is usually unnecessary.

Can I say 子供は早く伸びます and just leave out 髪?

No, that would sound strange. 伸びる needs something that “grows / stretches / lengthens” as its subject.

子供の髪は早く伸びます。 = Children’s hair grows quickly.
子供は早く伸びます。 by itself sounds like “Children grow quickly”, but usually for that idea you would use 成長する or 大きくなる, not 伸びる, so it feels unnatural.

If you want to say “Children grow quickly (physically / in general)” you’d say e.g.:

子供は成長が早いです。
子供はすぐ大きくなります。

What’s the difference between 髪 and 髪の毛?

Both refer to hair (on the head).

髪 (かみ) – shorter, more general word for hair (on the head).
髪の毛 (かみのけ) – literally “hair strands on the head”; can sound a bit more concrete/physical.

In everyday speech, both are fine:

子供の髪は早く伸びます。
子供の髪の毛は早く伸びます。

The meaning here is the same; 髪 is a bit more concise.

Why is the verb in polite form (伸びます) instead of plain form (伸びる)?

Japanese has different politeness levels:

Plain: 伸びる
Polite: 伸びます

You use 伸びます when:

talking to people you’re not close to
in class, textbooks, news, most writing for learners
in many formal or neutral contexts

You use 伸びる when:

speaking casually with friends/family
writing informal messages

So:

子供の髪は早く伸びます。 – neutral/polite
子供の髪は早く伸びる。 – casual/plain

The grammar is the same; only the politeness level changes.