After you unlock the Resident Services Building, you can ask Tom Nook about infrastructure, then choose to build a bridge.
Your First Bridge
If you haven't yet upgraded to the Resident Services Building, Tom Nook will give you a bridge building kit after you complete enough tasks for him. To build this first bridge, find a place where the riverbanks are the same distance away from each other and have smooth edges (no bumps sticking out into the water).
How to Build More Bridges
After you upgrade to the Resident Services Building, go inside and talk to Tom Nook. Ask him about infrastructure, then ask about building bridges and inclines. Choose bridges, and Tom Nook will show you the list of bridges that you can build. You need to have a free space in your inventory for the bridge building kit. You don't need to pay up front, so don't worry if you can't afford it right away.
You can have a maximum of 8 bridges on your island. If you want to build a new bridge but you already have 8, you have to talk to Tom Nook to destroy an existing bridge.
Bridges can vary in length depending on how far away the land on the other side is. The length can be 3, 4, or 5 spaces long. Bridges all require the same width: 4 spaces wide. The finished bridge will have room for two people to walk side-by-side.
Prepare the Land
If you have unlocked the Island Designer App, you can use the water terraforming tool to add or remove water. To prepare the land for a bridge, you need to make sure that the edges of the riverbank are four spaces wide where you want to build the bridge, with no bumps or jagged edges, and make sure that the distance from one riverbank to the other is is either 3, 4, or 5 squares of water, where you want to build the bridge. Bridges have to be a certain distance away from buildings, so make sure you are not trying to build a bridge too close to any structures.
Use the Bridge-Building Kit
Once you are ready, stand on one of the riverbanks and face the direction where you want the bridge to be. Use the bridge building kit, and if there is not enough room to build a bridge there, there will be a message explaining what is wrong. If there is enough room, you can choose to imagine what the bridge will look like. This will show you a preview of what the finish bridge would look like if you built it there. This will also show if any flowers or trees will be destroyed by building the bridge. If you are unsure, you can cancel and try something different if you wish. When you are ready, you can choose to build the bridge.
Pay Lloid and Wait a Day
After you place the bridge-building kit, Lloid the Gyroid will be there. You need to donate bells to him until the construction fee is paid in full. There is no time limit to paying the fee. After the fee is paid in full, the bridge will be built the next day.
Attend the Ceremony
After the bridge has been built, talk to Tom Nook to attend the ceremony. You and your residents, along with Tom Nook and Isabelle, will go to the bridge for a brief ceremony to celebrate its completion.