Q: i phone 5,  how can I monitor my childs use on internet, you tube, messaging and email?

You can set restrictions on each of those things that requires a passcode to access them but thats about it. I'm not aware of any kind of Net Nanny for the iPhone.

As already mentioned, restriction settings are the only way to control the usage of the phone.

This is also covered in the user guide iOS6 on page 137:

Restrictions

You can set restrictions for some apps and for purchased content. For example, parents can restrict explicit music from being seen on playlists, or prevent the installation of apps.

Turn on restrictions: Go to Settings > General > Restrictions, then tap Enable Restrictions. You’ll be asked to define a restrictions passcode that’s necessary in order to change the settings you make. This is distinct from the passcode for unlocking iPhone.

Important: If you forget your restrictions passcode, you must restore the iPhone software. See Updating and restoring iPhone software on page 152.

 

You can set restrictions for the following apps:

Safari

Camera (and apps that use the camera)

FaceTime

iTunes Store

iBookstore

Siri (including voice command and dictation)

 

You can also restrict the following:

Installing Apps: The App Store is disabled and its icon is removed from the Home screen. You cannot install apps on iPhone.

Deleting Apps: You cannot delete apps from iPhone. doesn’t appear on app icons when you’re customizing the Home screen.

Explicit Language: Siri attempts to replace explicit words you speak by replacing them with asterisks and beep sounds

Privacy: The current privacy settings for Location Services, Contacts, Calendars, Reminders, Photos, Bluetooth Sharing, Twitter, and Facebook can each be locked.

Accounts: The current Mail, Contacts, Calendar settings are locked. You cannot add, modify, or delete accounts. You also cannot modify iCloud settings.

Find My Friends: The current Find My Friends settings are locked. This option is available when the Find My Friends app is installed.

Volume Limit: The current sound volume limit setting is locked.

In-App Purchases: When In-App Purchases is turned off, you can’t purchase additional content or functionality for apps you download from the App Store.

Require Passwords: Requires you to enter your Apple ID for in-app purchases after the time period you specify.

Content Restrictions: Tap Ratings For, then select a country from the list. Then set restrictions for music, podcasts, movies, TV shows, and apps. Content that doesn’t meet the rating you select won’t appear on iPhone.

Multiplayer Games: When Multiplayer Games is off, you can’t request a match, send or receive invitations to play games, or add friends in Game Center.

Adding Friends: When Adding Friends is off, you can’t make or receive friend requests in Game Center. If Multiplayer Games is turned on, you can continue to play with existing friends.

I purchased an i-phone for my daughter a little over a month ago.  I did not realize at the time that none of the parent monitoring apps out on the market would work on the phone.  These applications work fine on Androids and allow you to limit when and how much time your child can use the phone and easily restrict access to inappropriate content and how long the child can play games and check on text messages for the security of your child, all remotely.  I am appalled that the i-phone reduces those applications to uselessness when installed and that you are stuck with limitations of the "Restrictions" internal to the i phone itself.  I am very unhappy with my purchase and the store only allows you 2 weeks to figure out the complexities of the phone and then you are stuck with it.  I do not recommend this phone for use by children if you as a parent wish to apply your parental prerogative.