Drill out the holes being careful to not go through front, fill holes by gluing in dowels the same size as holes. Cut off flush and drill pilot holes for new screws. This will give you solid new material for screws and will not be seen.

[i]If this is a duplicate I apologize, but my computer went off line as I submitted it, so I am not sure it went through.[/i]

Just a suggestion, you might want to try using expanding plastic or maybe metal wall anchors in the screw holes. Read [url]http://www.naturalhandyman.com/iip/inffastener/infanchor/infanchor.html[/url] about a quarter of the way down the page Plastic (and other)Expansion Anchors. I know that isn’t their purpose, but they are cheap and it can’t hurt to try.
Otherwise larger screws is the only other alternative I can think of – and eventually the same problem will probably recur.