  Let $s$ be an integer, $f=f(n)$ a function, and $H$ a graph. Define the
Ramsey-Tur\'an number $RT_s(n,H, f)$ as the maximum number of edges in an
$H$-free graph $G$ of order $n$ with $\alpha_s(G) < f$, where $\alpha_s(G)$ is
the maximum number of vertices in a $K_s$-free induced subgraph of $G$. The
Ramsey-Tur\'an number attracted a considerable amount of attention and has been
mainly studied for $f$ not too much smaller than $n$. In this paper we consider
$RT_s(n,K_t, n^{\delta})$ for fixed $\delta<1$. We show that for an arbitrarily
small $\varepsilon>0$ and $1/2<\delta< 1$, $RT_s(n,K_{s+1}, n^{\delta}) =
\Omega(n^{1+\delta-\varepsilon})$ for all sufficiently large $s$. This is
nearly optimal, since a trivial upper bound yields $RT_s(n,K_{s+1}, n^{\delta})
= O(n^{1+\delta})$. Furthermore, the range of $\delta$ is as large as possible.
We also consider more general cases and find bounds on
$RT_s(n,K_{s+r},n^{\delta})$ for fixed $r\ge2$. Finally, we discuss a phase
transition of $RT_s(n, K_{2s+1}, f)$ extending some recent result of Balogh, Hu
and Simonovits.
