Actually got the same problem,
Starting immediately after Mojave's installation.
Open safari, search from address bar: Apple ca
Google search appears 2 seconds before getting superseding with bing search screen and results.
.
Called Apple support, they made me install Malwarebytes, which identified 7 "major" threats, wow impressed as never ever scan my MAC for virus/spyware... they all find on windows hundreds of error, without fixing the problem...
.
So from chat been confirm to reboot issus should be resolved... lost chat.. 1:30 in morning so... reboot and guess what same problem.
Searching on my own:
So, safari preferences => search => if you select yahoo search and try the search all works through yahoo search.
If you select one of your favorite link, no issue, all work fine...
=> guess what the issue comes from internet explorer from your Mac windows...
Do you have internet explorer on your Mac, from parallels desktop, boot camp or else?
=>I have parallels desktop with windows 10 on the Mac book pro. As soon I updated to Mojave got the search issue.
=> to fix issue:
windows, start internet explorer 11 (default), from internet explorer options, remove your internet explorer security "lock mode"
Reboot as instructed windows, search for "google search engine". Click on the right IE search box, the arrow down and the click add . You will get requested to add google as search engine.
Select yes.
.
Once done, go to IE options manage extension, search engine, and check if you have bing and google as search engine.
Bing being #1 and default
Google being #2.
I have set google as default and then deleted
Bing as search engine.
Closed IE, and probably rebooted the windows again....
Guess what...
Safari issue got resolved by magic!!!
Hope it helps...