  Let $F$ be a family of subsets of $\{1,\ldots,n\}$. We say that $F$ is
$P$-free if the inclusion order on $F$ does not contain $P$ as an induced
subposet. The \emph{Tur\'an function} of $P$, denoted $\pi^*(n,P)$, is the
maximum size of a $P$-free family of subsets of $\{1,\ldots,n\}$. We show that
$\pi^*(n,P) \le (4r + O(\sqrt{r}))\binom{n}{n/2}$ if $P$ is an $r$-element
poset of height at most $2$. We also show that $\pi^*(n,S_r) =
(r+O(\sqrt{r}))\binom{n}{n/2}$ where $S_r$ is the standard example on $2r$
elements, and that $\pi^*(n,B_2) \le (2.583+o(1))\binom{n}{n/2}$, where $B_2$
is the $2$-dimensional Boolean lattice.
