Before revision:  There is C such that every 2-dimensional simplicial complex having n vertices and embeddable into \mathbb R^4 contains less than Cn^{ 5 /3} simplices of dimension 2.
After revision:  There is C such that every 2-dimensional simplicial complex having n vertices and embeddable into \mathbb R^4 contains less than Cn^{ 8 /3} simplices of dimension 2.