"search the web" bar on top of screen

Installing the "Vuze" bittorrent client from its website (vuze dot com) will maliciously modify Safari's Preferences and install a bunch of junk including the "Search the Web" browser hijack extension you seek to eradicate.

Removing it is simple and does not require that you download or install anything else. You certainly don't have to purchase anything or pay anyone to get rid of it for you. To do that would just be encouraging those seeking to capitalize on the misery you find yourself in.

Follow the procedure described below. Please be advised it is applicable only in this particular instance and as of this writing, because the distributors of the Vuze installer can change it at any time.

It's always a good idea to back up your Mac prior to making any changes to its file system, even if everything about it were working perfectly well. To learn how to use Time Machine please read How to use Time Machine to back up or restore your Mac - Apple Support.

If you see a "magnifying glass" icon in your Mac's menu extra area resembling the screenshot below, select "Quit":

• Open Safari's Preferences... menu > Extensions, and uninstall the "Smart Search" Extension by clicking its Uninstall button. A dialog box will appear asking you to confirm that's what you want to do. Do that by clicking its Uninstall button. That troublesome Extension will be gone.
• Click the General pane and review / reset your desired Home Page.
• Click the Search pane and review / reset your desired Search engine.
• Then: Return to the Finder and navigate to the following folder by selecting the entire line below and pasting it into the Finder's Go menu > Go To Folder... field:

~/Library/LaunchAgents/

A Finder window will open. In it, locate the following file and drag it to the Trash:

com.unionsoftwareonline.AppMonitor.plist

Repeat the Finder's Go > Go To Folder... with the following:

~/Library/Application Support

Locate the folder named "AppCommon" and drag it to the Trash.

(Optional) In the same manner as the above steps, locate the following files and drag them to the Trash:

/Library/Preferences/com.install4j.installations.plist

~/Library/Preferences/com.install4j.installations.plist

~/Library/Preferences/com.browserio.webexplorer.plist

• Finally: Open your Mac's Applications folder by selecting the Finder's Go menu > Applications. Locate the app named "WebExplorer" and drag it to the Trash.
  • Applications is also where you will find the Vuze app, should you wish to discard it too.
  • There will be a hidden folder in Applications with the title install4j. It's probably more trouble than it's worth to delete it, but you can do that if you are motivated to do so and know how to reveal hidden files and folders.
• (Optional) Empty the Trash.
• Log out or restart your Mac.

Please take note that the "Search the Web" modification is an example of adware that you agreed to install on your Mac—despite the fact you may not have fully appreciated the consequences of doing so. If you had taken the time to read the dialog boxes and various "terms and conditions" that accompany its installation, you would have learned that it would install the "Search the Web" browser hijack that redirects your search queries to a customized Yahoo! Search field, most likely for the sole purpose of directing clickthrough revenue to whatever sleazebags are promoting its installation. This is the sole reason for adware's existence: to take money from you, one way or another. The bulleted paragraph at the top of this post is applicable. Go back and read it again.

There is no one-size-fits-all magical cure-all to preventing or getting rid of such things, because a Mac still remains a more or less general purpose computing appliance yours to use or abuse as you see fit. The only effective defense against the threat of installing something that you don't want is to think before you click. Read the dialog boxes that accompany the software you're installing. Even when installing completely legitimate and beneficial software, it's your responsibility to know what it is, why you want it, and how to get rid of it if you decide you don't want it any more. For more on that subject please read Effective defenses against malware and other threats.