The action that improves the steering at a higher cornering speed is called "counter steering". Please practice it on a quiet mini roundabout (without your kid, of course), Enter the circus

at a higher speed than usual. Move the
outer pedal downwards then set your foot
firmly down on the pedal (the inner pedal will be up). Now, instead of turning the handlebars towards the turn direction, please
delicately push the inner bar-end
forward. This will make the bike lean towards the corner. The firmer you push the outer pedal down, the tighter the turn it will be. Although is sounds counter-intuitively, it is the technique used by both motor-bikers as well as advanced cyclists. It is because bikes turn by leaning towards the corner not by turning the handlebars at speed a little bit higher than just crawling

Cornering at a higher speed will let you maintain the momentum without pedalling!