My first home network Cat 5e, fiber 1 gig internet
Although it is true that Ethernet is bidirectional, it is still necessary that the packets from your fiber bridge reach the router before going to your other devices (and visa versa). This is because one of the functions of your router is to take the devices on your network and represent them to the outside Internet as a single device (actually multiple IP addresses as a single) using a protocol called NAT. Your network provider is probably expecting to see a single device (IP address) only and the router “fools” it.

Now because a single physical network can carry multiple logical networks through IP subnetting, it is possible to use a single cable between your garage and house and still run the packets to the router first. However, since this requires proper configuration of your local network and could require reconfiguration in the future if your provider’s configuration changes, I don’t recommend it to someone without knowledge of networking. It also requires an addition unmanaged switch at the router.

Assuming you need the router in the house, due to WiFi range limitations, I’d suggest that you run two CAT5E (or better) cables between the garage and house: one from your fiber bridge to the router WAN/Internet port and one back from a router LAN port to the switch.