Deixa que eu vou primeiro!

I am so confused about this.

not a native speaker but id think it would be ‘deixa que eu vá primeiro’ or ‘deixa eu ir primeiro’ (present subjunctive or personal infinitive’)

The phrase is correct.
Deixa que eu vou primeiro! = Bags I go first!
What @jamesp999 said, is also right, but that would change the meaning.
Deixa que eu vá primeiro = Let me go first.

@adrianxu I’m an American English speaker, I just had to google what ‘bags’ meant. Does that mean ‘I call dibs on going first?’ (american slang).

Is the meaning stronger and less like a polite request with the present tense instead of the subjunctive?

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Yes it does.

Exactly, saying “deixa que eu vou primeiro” is very informal and depending on the situation even rude.