iOS: Kids these days watch a lot of YouTube videos, and if you're worried about your child watching something inappropriate you can use the Gube app instead of handing them unrestricted access to YouTube. Gube curates thousands of educational and fun videos for your child in your choice of four age categories.
A few months ago we covered Kido'z, a similar type of app that can run on Android as well as Chrome and Firefox browsers. Now iOS users have a similar app that packs even more punch than Kido'z as there are more videos, each divided into infant, toddler, preschool, and grade school categories. Also Gube is a one-time cost of $3.99 rather than $4.99/month for the plus version of Kido'z.
My only criticism of the app is that it cannot lock the home button from being used, but that is more of a limitation of iOS than the app. You can always purchase Bubcaps to cover the home button before handing the phone to your child.
Gube is available for iPhone, iPad, and iPod Touch for $3.99 in the iTunes App Store.