начал читать

Я читаю в течение часа перед сном.

Something is wrong.

is the Russian incorrect?

Недавно and в течение часа don't sound good next to one another.

And adding перед сном makes the sentence even more weird.

OK, what I was trying to say is “I’ve recently started reading for one hour before bed.“

I see, but the literal translation doesn't seem to work.
Let us think.

Я почитаю час перед сном.¿
Я начал читать, а через час лягу спать. ¿

The speaker was actually talking about a new routine

The fact that they used the progressive indicates that it is something that they’ve started doing and are actively doing regularly.

They’ve started dedicating one hour to reading every night before they go to sleep.

Unless this is Anna‘s sentence, written by her, with the intention of describing something that she does habitually. In this case, the progressive is incorrect.

Ah! Я недавно начала читать перед сном в течение часа. Я с недавних пор перед сном один час читаю. Теперь я перед сном (один) час/в течение часа читаю.
Ι think my first sentence looks like your translation.
For some reason, it can be confusing without context.

👍🏼 add it

something that they’ve started doing and are actively doing regularly

Why progressive then?
For regularly present simple is normally used.
If it's have started doing that's important, then why not present perfect (continuous)?

That’s a very good question. The progressive is not the only way to say this. You can also say “I’ve been reading“ And it also implies a new activity that you recently started.

I have a feeling that Anna just means “I read for an hour“ and mistakenly use the progressive here.

So do I.