Use 'on the turn' when a situation is about to shift, signaling the start of that change. Her mood is on the turn; she seems happier now.
Pro Tip 2/3
Metaphorical Use
'On the turn' can be metaphorical for less tangible changes, like luck or emotions, not just physical ones. After the interview, I felt my fortune was on the turn.
Pro Tip 3/3
Neutral Tone
The phrase is neutral; it doesn't indicate if the change is positive or negative. With the weather on the turn, we could be in for sun or rain.