What is the best way of structuring an essay when comparing 4 poems?
I think having four poems is a bit much. You might want to limit it to 2. Not knowing what the poems are, it is a little difficult to answer. What are you comparing them based on? I would choose a common topic or theme, and look at how the poems treat it similarly and differently. Let's say the four poems are about love. One might be optimistic, the others pessimistic. Two might be about romantic love, while the others are about pessimistic love. Your overall thesis would be related to how love is explored in the four poems. If you have a choice, you might choose 4 by the same poet at different times in his life and take a chronological approach. For example, look at how the poet's view of love changes. In that case, your thesis might be that the poet became more pessimistic as he aged.
If I were you, I would post some more questions, with more specifics.