How can I access the Internet from an EC2 instance in a private subnet?

I have an EC2 instance in a private subnet within my VPC, and I'm using Session Manager to connect to it over port 443, which works fine. However, once I'm connected, I can't use 'wget' to download anything from the Internet. I've created a NAT gateway in the public subnet and set up a route table entry to redirect traffic from the private subnet to this NAT gateway, but it still doesn't work. I also tried creating a public NAT gateway to the private subnet and adding a default route, but I still can't access the Internet. Does anyone have any suggestions for solving this?