Hello there Kim,
I see you're noticing that your Safari search history is appearing on a family members device, so let's look into this so it's private.
For starters, are the two of you sharing the same Apple ID and iCloud login? If so, this would be why that information is syncing to their device. If you're wanting to disable this, you can disable Safari within the iCloud settings if you want to continue to use the same Apple ID for the devices. For help with this, use the following resource:
Set up and use iCloud Tabs
If you're wanting the other person to have their own Apple ID, you can still share purchases using Family Sharing but that would prevent personal information between the two of you from syncing to each others device. If this is for a child, you can also set up an Apple ID for them which can be used with Family Sharing to allow your information to remain private. If you'd like more information on Family Sharing, or setting up an Apple ID for a child, take a look at these resources:
Set up Family Sharing
Create an Apple ID for your child
I hope this is able to help.
Have a great day!