What bit would I use to make this cutout

As others have stated you need a pattern (4") and a pattern bit. If you have a circle cutter then make the cut on a drill press with the pattern clamped down. Then use carpet tape to glue the pattern to your board. Then use the plunge to route out the hole about 1/8" at a pass. You go around the circle with the pattern bit then route out the middle by just moving the router clockwise until all of the layer is cut. Your pattern must be tall enough to have the bearing contact the pattern during the first pass. Your pattern bit needs to be on the top of the router bit. A bottom bearing would not allow you to plunge into the pattern. Just remember if you have a 1/2" bit then never take more than 1/2" deep cut at a pass. If possible make shallower cuts multiple times. There is a lot of material to remove at one time 1/2" so make shallower passes to get a smooth bottom. If you have a larger bit like a 3/4" pattern bit do not make deep plunges because the torque of the router will make it hard to control. The example picture does not have a lot of surface for the router to sit on and your board needs to be secured or you will make a mess.

If you do not have a circle cutter then use a 4" hole cutter. You will get a rough edge. You can still use the pattern bit but you will need to plunge inside the 4" hole with a mortising bit to get a place to get the pattern bit in. Make sure your plunge hole is bigger then your pattern bit.

The suggestion about cutting a 1/2" piece, then cut the hole and glue on a bottom is a good one but the hole still has to be cut.

So if you only want to use the router you need a pattern. The pattern can be made with a hole saw or a circle cutting bit. There are no short cuts. You just keep getting sucked in further and further.

If you do not have a drill press then see if you can go to a neighbor or relative and use theirs. Hand drilling with a large bit and/or hole saw is dangerous. The torque will spin your project around and around. Secure everything to be safe.

If you are wondering how to secure your small work piece then you can cut some plywood strips and put your work piece on top of a piece of scrap plywood then screw the strips around the work piece. This gives you a larger work are for the router to sit on. You could also start with a larger work piece and after you get the hole routed then trim the work piece down to your required dimensions

Always look at a problem from different perspectives. If a carpenter only has a single hammer in his tool bag every problem looks like a nail.

I've just realised that you probably don't have the means to rout the template hole. Here is one very easy to make circle routing jig.

This is a set of guide bushings (I'm not particularly recommending these exact ones): https://www.amazon.ca/Shop-Fox-D311...=2025&creative=165953&creativeASIN=B0000DD4MN They can be used to follow templates too but they have a slight offset. They can also be used to slightly enlarge or make smaller pieces than a particular pattern by altering the size of bit and bushing. Harry Sinclair posted some excellent tutorials on using them years ago. They'll be somewhere on the home page.

A straight bit with a bearing on the shaft is referred to as a pattern bit. With a pattern bit you follow a template that is between your work piece and the router (above or below is relative to whether you are doing this handheld or on a router table). It does the same job as a flush trim bit with the bearing on the bottom with the difference being that the pattern is now attached to the other side of the work piece. On a router table I prefer using a flush trim bit as the top of that bit is the bearing. Using a pattern bit on a table has the cutter edges exposed instead making the flush trim bit safer to use.

For something that depth, I'd use a mortising bit with a top bearin. Cut the circle template in some (3/4 inch) MDF and perfect the edges with sandpaper wrapped around a large dowel, or on a sander. MDF dust is toxic and sticks inside the lungs, wear a mask and sand outside if you can.

Use double sided tape to stick the template to the target piece. A little carpet tape will do. Use a plunge base to gradually lower the bit into the work. I wouldn't use a guide bushing, unnedessarily complex to set up compared to a mortising bit with bearing. The mortising bit is made to give you a very flat bottom. Cut no more than 1/8th per pass. Gradually hollow out each layer before plunging to the next. Consider making the last pass just 1/16 inch so you don't risk any tear out.

First pix is of a half inch, Freud mortising bit and the cut it makes. Very smooth, flat bottom. Second pix is of a 3/4 inch mortising bit. Both are easily available at HD or online. You also use it for things like cutting a shallow mortise for a hinge--look at the hinges on a door to see how it's recessed.

I hope you're using a router with a half inch shank, the lateral forces involved would put a lot of stress on a 1/4 shank.

My half inch mortising bit got a real workout a couple of years ago when I replaced all the interior doors in our house. Former owner's dogs ruined the originals.