The limit is not your resolution. It is the tiny size of your camera sensor. The sensor is about 4.8x6.4mm or possibly 4.8x7.2mm. Despite the 20MP claim your camera is not really capturing 20MP of detail. It is closer to 3 or 4MP. I doubt that you would be able to see any difference between a 10 MP image interpolated up to 20 MP and the same image shot at 20MP. If your sensor was really capable of capturing 20 MP of data, then full frame cameras would have 500 MP sensors rather than the typical 50MP.
With a small sensor, to get to 8x10 inches each dimension is being enlarged at least 42 times. Even so most small sensor cameras will deliver good 8x10 images. Going bigger introduces a lot of variables, how much sky? How many fine details especially in the greens and blues? What ISO did your camera pick? You may not see the difference between an ISO of 200 and 1600 on a 13 inch laptop, but the bigger you print the more brutally obvious that difference will become. Given that your bar seems fairly low you may be able to take some images to 11x14 or even bigger, but not if there is a lot of fine detail or if the camera selected a high ISO. I've said it repeatedly, if you want to be able to count on prints larger than 8x10, you need a bigger sensor. Especially as you seem to expect the camera to do all of the work for you. Keep in mind that I have owned 6 small sensor cameras, one of which is still alive. I know their limits and love working with them. There is much to be said for a camera that can be tucked into pocket or carried in a purse or on a belt. They are fine for many photographers as long the photographer understands the cameras limits.
The next size up in sensors is the 1" sensor which is 8.8x13.2mm. These should easily produce excellent 16x24 prints as long as cropping is minimal. There are a handful of pocket sized cameras in this category, but they are more expensive than their small sensor counterparts.