Steep uncountable groups

We produce a simple group $G$ of cardinality ${\mathrm{\aleph }}_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y\subseteq G$ there exists a natural number $n_Y$ for which every element of $G$ may be expressed as a product of length at most $n_Y$ of elements in $Y^{\pm 1}$. In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group $G$ can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in $G$.