safeTube is an application that allows you to filter YouTube content that you find acceptable for your child.
YouTube comes with some basic content controls. Unfortunately, all this does is block content as voted either by YouTube members or YouTube itself. Even with these content protections, YouTube is a rich internet site, so if the first video found is acceptable, there is a chance that there are other videos whose icon previews or titles/comments you may find objectionable.
Additionally, there is no way to prevent your child from trying different searches for videos, and accidentally come across a video that you find objectionable.
On a Mac using OS X 10.5 or better, go to the parental controls for your child's account, and ensure YouTube is not a site they can visit on their browser. Then, install safeTube so that is may act as a window into YouTube.
On the iPhone, install the safeTube app from the app store, but ensure that YouTube is enabled via the Settings -> Restrictions menu. (due to Apple policies, YouTube must be enabled for safeTube to work.)
When launched, safeTube presents a list of allowed search terms. To enter a new YouTube search, a password must be entered (set up on first use of the application). Without that, no searches can be performed.
Once searches are logged, your child can click on one of the searches, and only videos that match that search term will be visible. They can select one of the videos, and it will play. When the video is finished, there is nothing else they can do - there are no suggestions that come up for related videos, for example. All your child can do is replay that video, or play other videos that matched the approved search terms.
Thus, your child cannot be led to other parts of YouTube, eventually leading to areas you do not find appropriate for them.
If your child finds a video they particularly like, they can mark it as a favorite, and as long as that video remains on YouTube, they can watch it over and over.
You can supply an endless list of searches - there is no limit. At any time, you (or your child) can remove a search item from the list, and they will no longer be able to view videos matching that search unless you re-allow it.