When woodworking plans call for cutting curves in thick stock, typically the tool of choice is the band saw. Band saws can be easily adjusted to accommodate most thicknesses of stock.
However, what tool should you choose when you need to make a cut out inside of a piece of stock (such as a cutout for a handle) where you can't get a continuous band saw blade into position to make the cut? Or what if you don't have access to a band saw?
The first thought would probably be to use a jigsaw (as shown above) to make the cut. However, jigsaw blades have a tendency to flex and bend, which can result in an angled cut, which can cause more problems to deal with.
Perhaps a better idea would be to cut a template out of much thinner material with your jigsaw (where the blade would be less likely to flex). Then, once the template is cut, attaching the template to the thick stock will allow for a router with a straight cutting bit and a bearing tip to cut the appropriate curves in the thick stock.
There are some drawbacks to this method. First, use your jigsaw to cut close to the actual curves to remove most of the excess stock. Then, make slow, deliberate cuts with your router, removing small amounts of stock at a time. Attempting to remove all of the remaining stock in one pass is likely to result in a burned router bit, so don't get aggressive.
Even if you're careful to work slowly, you may need more than one router bit to complete the cut, which can be expensive. Spiral bits will probably last longer than straight cutting bits.
There are some router-like "all-in-one" tools that might be a good choice for this type of task, but many of these tools may not be able to handle overly thick stock.
While I've never tried it, I do know of woodworkers who have used their jigsaw to make curved cuts in thick stock, but to do so, they placed two jigsaw blades in their jigsaw. While not all jigsaws can accommodate two blades simultaneously, it might be worth investigating. You'll want to go slow, but the second blade will help minimize blade flex.