Virtualbox can ping but can’t browse

VirtualBox handles NAT networking internally -- you don't need to configure any iptables rules. Remove your iptables configuration, and then test that the VM has outgoing connectivity. In order to create incoming connections to the VM you'll need to setup port forwarding.

So you are bridging the VM NIC with a 192.168.0.x ip to a LAN which runs 192.168.1.* ?

That ain't going to work of course.

When you bridge the VM NIC works "in parallel" to the hosts NIC on the same LAN.
That means that BOTH machines (host and guest) are in the SAME subnet and should use an ip-address in that range.

You can make it work with different ip-ranges, but you will have to re-configure the router in that case. It must have an ip-address in each subnet and route between the 2 L3 subnets on the same L2 LAN.
And both machines need to use that router (the routers ip-address for their own subnet) as their default gateway.

Please note that super-imposing 2 subnets on the same L2 LAN may cause some undesired side-effects. A lot of equipment won't like it. It may also cause some issues with your DHCP server (e.g. if it is a Microsoft DHCP server the superscope needs to be setup correctly).