널 잊으려고 밤새 달렸어

The line, explained

Picture the music video. Night, an empty bridge, a figure running with their hood up and the city lights smeared behind them. Korean lyrics are full of people doing something to stop thinking about someone else, and the grammar they use is -(으)려고, 'in order to'. This line has it in the second word. Everything after that is the effort; the first two words are the reason, and the reason is a person.

널 is 너를, 'you' as the object of the verb, squeezed into one syllable. Lyrics almost never write 너를 in full; the two-syllable version slows the beat. 잊다 is to forget, a verb with a consonant stem, so it takes -으려고 to become 잊으려고: 'in order to forget', 'trying to forget'. The subject is understood to be I. 널 잊으려고 is already a complete thought, 'to forget you', and a singer could stop there and let the music fill in what they did.

The rule for -(으)려고 is the same split you know from other endings: consonant stems take -으려고 (잊으려고, 먹으려고), vowel stems take -려고 (가려고, 보려고). One warning. You will see 잊을려고 and 갈려고 in comments and song captions, with an extra ㄹ in the middle. That is a very common spoken slip, not the standard form. Write it without the ㄹ and you will be both correct and understood everywhere.

밤새 means all night, through the night. 밤 is night, and 새 here is a shortening of 사이, the span between, so 밤새 is literally 'the whole span of the night'. 달렸어 is 달리다, to run, in the casual past tense; 리 plus 었어 contracts to 렸어. 밤새 달렸어 is 'I ran all night'. Put the purpose first and the action second and you have the natural Korean order: 널 잊으려고 밤새 달렸어, to forget you, I ran all night.

What the line does not say is whether it worked. -(으)려고 only reports the intention; the result is left to the listener and the next verse. That gap is the whole mood of the song. Say it with the stress on 잊으려고, the reason, and let 달렸어 come out tired: neol i-jeu-ryeo-go bam-sae dal-lyeo-sseo. Then try the same frame with the opposite goal: 널 보려고 밤새 달렸어, I ran all night to see you.