No, a leaking mask is not normal. It's possible that your mask doesn't fit properly, the strap is adjusted too tight, something (hair, hood, etc.) is preventing the mask skirt from sealing properly, or you are inadvertently exhaling through your nose.
I cannot say for certain why your snorkeling mask leaks at deeper depths but appears to work just fine in shallow water. Generally speaking, however, scuba masks are made with better materials than masks meant for topside activities. Scuba masks incorporate tempered glass and good-quality silicone skirts into their design. This is not necessarily true with snorkeling masks.
The sheer number of available masks on the market can be overwhelming to a novice diver. I would advise you to shop for a dive mask at a local dive shop. A knowledgeable dive shop employee and/or experienced diver should be able to help you select a mask that fits your face. Understand that just because a mask
seems to fit you in the shop, it doesn't necessarily mean that the mask will work underwater. The mere fact that a regulator is in your mouth could very well change the contours of your face (position of lower jaw, cheek shape, etc.). For this reason, it might be helpful to bring along your regulator mouthpiece while you try on masks in the store. When I had to replace my mask a few years ago, that's exactly what I did.
Finding the right mask for you is very much a trial-and-error process. Some brick-and-mortar dive shops and online retailers have very liberal return policies when it comes to a dive mask. If you can find such a shop, take advantage of this "personal fit guarantee." Diving simply isn't fun if your mask doesn't fit.
Once you find a mask that works for you, buy at least one more identical mask as a backup. There's nothing more frustrating than finding out that the
one mask that fits your face has been discontinued by the gear manufacturer. A good quality scuba mask should last several years if you take care of it.
For more info about finding the right mask for you,
read spectrum's essay on the subject. It contains a lot of great info.
Good luck!