I can think of at least three proofs:

1. The original cell decomposition proof of Szemerédi-Trotter

2. The proof via the Crossing Lemma given by Székely

3. The proof using the polynomial ham sandwich theorem

Adam Sheffer has an early draft of a book on incidence theory and the polynomial method, which mentions all three proofs.

I can think of at least three proofs:

1. The original cell decomposition proof of Szemerédi-Trotter

2. The proof via the Crossing Lemma given by Székely

3. The proof using the polynomial ham sandwich theorem

Adam Sheffer has an early draft of a book on incidence theory and the polynomial method, which mentions all three proofs.