i have exactly the same problem. in the meanwhile i have tried restoring my phone from an icloud backup. that was not successful. then i tried a restore from a ituens backup. also without a restore. as last resort i gave up my photos and my imessage history and setup the phone as a brand new device. safari then worked for few pages but after i have finished setting up all my apps and settings it did not work again.
i did some more research and came accross this useful link here: http://osxdaily.com/2015/03/10/having-safari-problems-with-ios-8-2-try-this/
in one of the comments somebody suggested to try the private browsing in mobile safari application. that seems to work for me. it is not a real solution to the problem but a workaround so you could still use safari and the favourites saved in icloud.
IHMO: i think it is a serious bug in ios 8.3 and apple should fix it asap so safari gets usuable again...