"She needs a large bag."
Translation:Ей нужна большая сумка.
You have three problem points in your sentence.
Bag is сумка, which is feminine singular.
Большой must decline to match the gender and number of what it describes, and so it becomes большая (feminine singular) to correspond with сумка (feminine singular).
Нужен, нужна, нужно, and нужны are short-form adjectives which must match the gender and number of what they describe. Since сумка is a feminine singular, you want to use нужна, not нужно. Нужен would be used for masculine singular items. Нужно would be used for neuter singular items. Нужны is used for all plural nouns regardless of gender; you would use that when you wanted to say Ей нужны большие сумки, she needs large bags, or Ей нужны яблоки, she needs apples, etc.
I know what you mean but I had the impression that it is also possible to formulate the sentence as Ей нужно (as an adverb) чего (genitive). Or is it another situation that I got mislead by?
I do not know about any rules involving нужно + genitive, so I was coming at it from another point of view. I am sorry, I cannot help you - we will need a native speaker in this case. :)