Suppose there is an upper bound D on the diameter of the polygons.
Consider nested squares where the outer square has a margin of D
on all sides of the inner square.
Let R be the ratio of the area of outer square to the area
of the inner square.
R approaches one as the inner square becomes large.
Choose the inner square large enough that R < 1.5.
Let X be the area (and number of lattice points)
of the inner square.
The number of polygons that intersect the inner square is at least X,
each has area 1.5, and each is contained entirely within the outer square.
So the area of the outer square is at least 1.5X, contradiction.