Obstructions to embedding singular curves in toric varieties

For every integer $d\ge 2$, we show that there exists an irreducible, reduced curve which embeds in ${\mathbb{P}}^d$ but in no projective normal toric variety of dimension less than $d$. In particular, there exist reduced irreducible curves that embed in ${\mathbb{P}}^d$ but do not embed in any weighted projective space of dimension less than $d$.