No, it doesn't.
You have to actively go and look for it yourself, the browser is just a means of transport. It would be different if the browser had a menu that said "download illegal content here". Like FTP or a newsreader isn't illegal, it's the act of you accessing illegal content that is breaking the law. Same as with the Pirate Bay website. It's not the browser that is offering illegal content, it's the torrent site of the Pirate Bay.
disclaimer: I know the the laws are different in every country, but we have to comply with Dutch law.