Once the text in a handout becomes long enough to scroll the styles palette out of sight it becomes impossible to apply styles. If you select and scroll up to find the palette, scrolling loses you selection. If you choose a style and scroll down to resume typing scrolling loses you choice on the palette. If you select a style and then just type nothing happens, because scrolling loses you the text insertion point.
The only thing to do is to prepare handout text off-line and import it fully formatted.