My kids borrow my iPhone all the time to play music, watch YouTube videos and Google stuff. I've already had a few close calls where a shipping notification from the Google search app nearly exposed a package that allegedly will be hand-delivered from the North Pole in a couple of weeks.
The Google search app -- on iPhone and Android -- pulls in shipping information from Gmail when I receive a shipping confirmation email from Amazon or whomever. Theses notifications show up in three places on my iPhone: the lock screen, the Notification Center and as cards on the start page of the Google search app itself.
Say 'no' to Lock Screen and Notification Center alerts
Alerts on the lock screen are the chief concern here. Without knowing your passcode, your kids can accidently spy shipping notifications on your locked phone.
Go to Settings > Notifications and scroll down to the line for Google. At the top of the screen, you can turn off the toggle switch for Allow Notifications, which disables all notifications from the app. If this nuclear option is too much for you, then you can keep allowing notifications but turn off the toggle switches for Show on Lock Screen and Show in Notification Center. This allows you to receive alerts when your iPhone is unlocked.
Don't deal me in, Google
If your kids use your iPhone to Google answers to their many questions, then you will want to disable shipping notifications that show up as cards on the Google search app. To do so, launch the app, find a card with a shipping notification and tap the triple-dot button on its right edge. From the menu that pops up, tap Never show package tracking.
To make doubly sure that all shipping notification cards are turned off, tap that triple-dot button again on any card and tap Customize the feed. Next, tap on Gmail and then tap to turn off the toggle switch for Package tracking. (I thought that when I selected Never show package tracking, the Package tracking toggle switch would then be turned off, but it wasn't.)