Barnen får springa på gräsmattan, men de får inte gå genom busken.

Because barn is an irregular noun.

• ett barn = a child
• barn = children
• barnen = the children

So the plural definite form is barnen, not barnarna.

This is a very common Swedish pattern with some neuter nouns: the singular and plural can look the same, and the ending -en marks the in the plural.

Here får means may / are allowed to / can in the sense of permission.

The verb is få, which has several meanings depending on context. For example:

• Jag får en bok. = I get a book.
• Jag får gå nu. = I may / am allowed to go now.

In your sentence, får springa and får inte gå are about permission, not receiving something.

Because after få used as a modal-like verb, Swedish normally uses the infinitive without att.

So:

• får springa = may run
• får inte gå = may not walk/go

This is similar to English modal verbs:

• can run, not can to run
• may go, not may to go

In a main clause, Swedish usually puts the negation inte after the finite verb.

So:

• de får inte gå = they may not go

Here the finite verb is får, so inte comes after it.

Compare:

• De springer inte. = They are not running.
• Hon kan inte komma. = She cannot come.

This is a very important Swedish word-order rule.

De means they, and it refers back to barnen.

So the sentence structure is:

• Barnen får springa på gräsmattan = The children may run on the lawn
• men de får inte gå genom busken = but they may not walk through the bush

Just like in English, Swedish often uses a pronoun instead of repeating the noun.

Because på is used for being or moving on a surface.

A lawn is thought of as a surface area, so Swedish says:

• på gräsmattan = on the lawn

Using i would suggest being inside something, which does not fit a lawn.

This is similar to:

• på golvet = on the floor
• på stranden = on the beach

Because Swedish often puts the definite article at the end of the noun instead of using a separate word like English the.

• gräsmatta = lawn

• gräsmattan = the lawn

• buske = bush
• busken = the bush

So the -n ending here is doing the job of English the.

• springa = to run
• gå = to walk / go

In this sentence:

• springa på gräsmattan = run on the lawn
• gå genom busken = walk/go through the bush

Swedish gå is broader than English walk in some contexts. It can sometimes translate as go, but here the idea is physical movement on foot.

Genom means through.

It usually describes movement from one side of something to the other side, or passing through an area/object.

Examples:

• gå genom dörren = go through the door
• köra genom tunneln = drive through the tunnel
• gå genom busken = go through the bush

So in this sentence, the children are not allowed to pass through the bush.

Yes, Swedish could use either singular or plural depending on the situation.

• genom busken = through the bush
• genom buskarna = through the bushes

Your sentence uses the singular busken, which suggests one specific bush. If the real situation involved a group of bushes or shrubbery, Swedish might instead say buskarna.

So the grammar is straightforward here: singular definite noun = busken.

Yes. In this sentence, men means but.

It connects two contrasting ideas:

• the children are allowed to do one thing
• they are not allowed to do another thing

So:

• Barnen får springa på gräsmattan, men de får inte gå genom busken.
• The children may run on the lawn, but they may not go through the bush.

They are both in the definite singular.

gräsmatta

• en gräsmatta = a lawn
• gräsmattan = the lawn

buske

• en buske = a bush
• busken = the bush

Both are en-words in Swedish, so in the singular definite form they usually take -n.

The sentence follows normal Swedish main-clause word order:

1. subject
2. finite verb
3. other elements

First clause:

• Barnen = subject
• får = finite verb
• springa på gräsmattan = rest of clause

Second clause:

• de = subject
• får = finite verb
• inte = negation
• gå genom busken = rest of clause

So the pattern is very typical Swedish:

• Subject + finite verb + inte + infinitive ...

That makes this sentence a good example of standard Swedish sentence structure.