In YouTube, scroll all the way to the bottom and click on the "Safety". You can also lock on this feature, which means the only person that can unlock it is the owner of the account.
In Google when you search, both images or web content, you should see a "Safe Search" is on box appear on the top right. If you don't see this box, you probably don't have it turned on. Click into Search Settings to check the appropriate box, then be sure to save your updated settings.
Google also allows you to lock your safe search to the On position. If safe search is locked on, balloons will appear in the upper right hand corner for a quick glance check.
