The browser may have had a close-encounter with some adware or other exploit
due to visiting sites and clicking on content, or at some previous time, having
visited a page that may have offered free bad software. adware malware & etc.
•Remove unwanted adware that displays pop-up ads and graphics on your Mac - Apple Support
If you'd installed any of the items mentioned in the above linked page, or have symptoms
of their presence, they need to be addressed. You can start Safari in a mode where the
extensions or add-ons are turned off, then you can see about removing them. And others.
You could see a site by name of TheSafeMac and read the sections on Tech Guides
and note there are three, for adware malware & performance. Also there is a link in the
right side of the page to get the Adware Medic adware removal tool. Or use the manual
methods as described in the Adware Removal Guide in the main page content below:
•The Safe Mac - Tech Guides:
http://www.thesafemac.com/tech-guides/
You could get & try the etrecheck report creation tool to see what may be acting up
in your computer, in a superficial way, and re-post the report to a discussions area.
Etrecheck: http://www.etresoft.com/etrecheck_story
This probably is the nature of the topic you have posted. Where you can't change the
search engine in safari and other oddities have occurred that were not original features.
If it somehow that happens to not be the case (rare but possible) the method to choose
a different default browser in general, is to use Safari Preferences and choose another
such as Firefox, as the default. Then the other one will launch as system default. This
action would not remove adware, malware, or other maladies, nor prevent them.
In any event...
Good luck & happy computing! 🙂