Convert all youtube link to embed

EDIT

After reviewing your edit, you may want to try following These Instructions

To embed a video or another object into a post or page, place its URL into the content area. Make sure the URL is on its own line and not hyperlinked (clickable when viewing the post).

It sounds like you would want to use wp_get_oembed() – The examples look like this:

Retrieve the embed code for a URL from a supported oEmbed provider:

<?php $embed_code = wp_oembed_get('http://www.youtube.com/watch?v=tkEvSjdDfiA'); ?>

Retrieve the embed code for a URL from a supported oEmbed provider – with width argument:

<?php $embed_code = wp_oembed_get('http://www.youtube.com/watch?v=tkEvSjdDfiA', array('width'=>400)); ?>

If that doesn’t work you could probably apply ‘the_content’ filter:

apply_filters( 'the_content', 'http://www.youtube.com/watch?v=tkEvSjdDfiA' );

But that filter is probably overkill for what you’re trying to achieve.