I don't understand how you have 'various search engines' on tool bars. One answer might be to open your favourite search engines and add them to 'favourites' (the green and yellow icon top left) and then access them from there.
It sounds as though you might have put them in 'favouites' and then moved them into the 'Links' folder which would make them appear separately on your (Links) tool bar.
To take it a step further, in Favourites you could create a new folder, to put the search engines in, and then move that folder into Links. That way you would have just the folder appear on your (Links) tool bar and be able to access its contents from there as you choose.