  In [Bak] the first author proved that for any $\beta\in (1,\beta_{KL})$ every
$x\in(0,\frac{1}{\beta-1})$ has a simply normal $\beta$-expansion, where
$\beta_{KL}\approx 1.78723$ is the Komornik-Loreti constant. This result is
complemented by an observation made in [JSS], where it was shown that whenever
$\beta\in (\beta_T, 2]$ there exists an $x\in(0,\frac{1}{\beta-1})$ with a
unique $\beta$-expansion, and this expansion is not simply normal. Here
$\beta_T\approx 1.80194$ is the unique zero in $(1,2]$ of the polynomial
$x^3-x^2-2x+1$. This leaves a gap in our understanding within the interval
$[\beta_{KL}, \beta_T]$. In this paper we fill this gap and prove that for any
$\beta\in (1,\beta_T],$ every $x\in(0,\frac{1}{\beta-1})$ has a simply normal
$\beta$-expansion. For completion, we provide a proof that for any
$\beta\in(1,2)$, Lebesgue almost every $x$ has a simply normal
$\beta$-expansion. We also give examples of $x$ with multiple
$\beta$-expansions, none of which are simply normal.
  Our proofs rely on ideas from combinatorics on words and dynamical systems.
