There's a combination of things that need to be done in order for this to work, it's not an easy one.
Your typical, default NAT works like this:
+-------+ +-------+
| Guest |--> | Host |--> Router -> Internet
+-------+ +-------+
Your Guest *has* to go through the Host, it doesn't work the other way round.
Your Bridged mode means that the Guest is using a Host's NIC to access the Router:
+-------+
| Host |---+-> Router -> Internet
+-------+ |
+-------+ |
| Guest |---+
+-------+
What you're looking for is to "cut" the Host-to-Router connection (red dashes in the diagram above) and establish a Host-to-Guest connection, something that's going to be running as a gateway to your Host:
+-------+
| Host | +-> Router -> Internet
+---+---+ |
| |
+---+---+ |
| Guest |---+
+-------+
That gateway from what I've seen proposed time and time again is called pfSense and it's running in your Guest. Search for "
pfSense site:forums.virtualbox.org" and read the instructions in the topics that you'll find.
We should make a HowTo for this someday...