Most browsers like Google Chrome and Firefox will search a we page from the current selection onward. It seems Safari does not support this.
Instead it searches from the beginning of the webpage.
How can I change this behaviour?
You can easily search from the point of selection using the following approach.
Immediately after selecting the string to search, use the keyboard shortcut Command + E to record the search string into find buffer.
Now press the keyboard shortcut Command + G to search from the selection point in the forward direction. To search backward, use the keyboard shortcut Command + Shift + G.
This works in most other apps apart from Safari.