Hun sier æsj på grunn av lukten.

Yes. Æsj is a very common Norwegian interjection, used when something seems disgusting, gross, or unpleasant. It works a lot like yuck, ew, or ugh in English.

In this sentence, Hun sier æsj means that she reacts by saying æsj.

Sier means says, so the sentence describes what she actually says out loud.

• Hun sier æsj = she says yuck
• Hun tenker ... = she thinks ...
• Hun føler ... = she feels ...

Because æsj is something spoken as a reaction, sier is the natural choice here.

Because æsj is not a full clause. It is just an interjection.

You use at after sier when what follows is a full statement:

• Hun sier at hun er trøtt. = She says that she is tired.

But here, she is just saying a single reaction word:

• Hun sier æsj.

So at is not needed.

På grunn av means because of.

It is used before a noun phrase, not a full clause.

In this sentence:

• på grunn av lukten = because of the smell

Compare:

• på grunn av regnet = because of the rain
• på grunn av sykdom = because of illness

If you want to say because followed by a full clause, Norwegian usually uses fordi instead:

• Hun sier æsj fordi det lukter vondt. = She says yuck because it smells bad.

Because fordi normally introduces a clause with a verb, while på grunn av is followed by a noun phrase.

So:

• på grunn av lukten = because of the smell
  
  • lukten is a noun phrase
• fordi det lukter vondt = because it smells bad
  
  • this is a clause with the verb lukter

That is the key difference.

Lukten is the definite form of lukt.

• lukt = smell, a smell
• lukten = the smell

Norwegian often uses the definite form when talking about a specific thing already understood from the situation. Here, it means the smell that is causing her reaction.

So:

• på grunn av lukten = because of the smell

Yes, in Bokmål both lukten and lukta can be used.

• lukten is often felt to be a bit more standard or neutral in writing
• lukta is also very common and often sounds more everyday or colloquial

So these are both possible:

• Hun sier æsj på grunn av lukten.
• Hun sier æsj på grunn av lukta.

The sentence follows normal Norwegian main-clause word order:

• Hun = subject
• sier = verb
• æsj = what she says
• på grunn av lukten = reason phrase

So the structure is roughly:

subject + verb + rest of sentence

That is very similar to English here:

• She says yuck because of the smell.

Æsj is pronounced roughly like a short, sharp disgust sound.

A helpful approximation for English speakers is something like:

• esh or ash, but with the Norwegian vowel æ

The sj sound is like the sh sound in English ship.

So the whole word sounds approximately like esh with a disgusted tone.

By itself, av often means of, from, or by, depending on the sentence. But in på grunn av, you should learn the whole expression as one unit.

So it is best to think of:

• på grunn av = because of

rather than trying to translate each individual word separately every time.

Yes, that is a very natural translation.

In English, you might also say:

• She goes yuck
• She says ew
• She says yuck because of the smell

So the literal translation works well here.

Yes, it is natural and understandable. It sounds like normal everyday Norwegian.

A native speaker might also say similar things such as:

• Hun sier æsj fordi det lukter vondt.
• Hun sier æsj på grunn av den lukten.
• Hun sier æsj når hun kjenner lukten.

But your sentence is perfectly fine as it is.