The famous Szemerédi-Trotter theorem states that any arrangement of n points and n lines in the plane determines O(n4/3) incidences, and this bound is tight. Al-though there are several proofs for the Szemerédi-Trotter theorem, our knowledge of the structure of the point-line arrangements maximizing the number of incidences is severely lacking. In this talk, we present some Turán-type results for point-line incidences. Let L1 and L2 be two sets of t lines in the plane and let P = {l1 ϵ l2 : l1 ϵ L1,l2 ϵ L2} be the set of intersection points between L1 and L2. We say that (P,L1 ϵ L2) forms a natural t × t grid if |P| = t2, and conv(P) does not contain the intersection point of some two lines in Li, for i = 1,2. For fixed t > 1, we show that any arrangement of n points and n lines in the plane that does not contain a natural t×t grid determines O(n-43ϵ) incidences, where ϵ = ϵ(t). We also provide a con-struction of n points and n lines in the plane that does not contain small cycles and determines superlinear number of incidences. This is joint work with Andrew Suk and Jacques Verstraete.