I can think of at least three proofs:

1. [The original cell decomposition proof of Szemerédi-Trotter][1]

2. [The proof via the Crossing Lemma given by Székely][2]

3. [The proof using the polynomial ham sandwich theorem][3]

[Adam Sheffer has an early draft of a book][4] on incidence theory and the polynomial method, which mentions all three proofs.


  [1]: https://link.springer.com/article/10.1007%2FBF02579194
  [2]: https://www.cs.umd.edu/~gasarch/TOPICS/erdos_dist/szekely.pdf
  [3]: http://yufeizhao.com/pm16/matousek-szt-poly.pdf
  [4]: http://faculty.baruch.cuny.edu/ASheffer/000book.pdf