If you want to hide the “Conversations” options from the dropdown menu that is shown after the Search box, the easiest way is to CSS hide it. Set the following class to “display:none”.
Then you want to remove the same option from the Search center page (results.aspx) also.
Edit the Search page (in the Search center)
Edit the Search Navigation web part
Click on “Search Settings Page”
Select “Conversations” and “Delete”