Those fears are categorized as survival reactions, or SRs. Perfectly normal - you can to learn why you have them, and how to control them.
Leaning is how a bike turns - which is to say it falls. To fully illustrate this counter steering needs to be introduced into the conversation. Counter steering is how a bike steers, which is different than three and four wheel vehicles. Turn the bars to the left and the bike falls and turns to the right. Turn the bars to the right and the bike falls and turns to the left. Well into the 1970s and 80s people didn't believe this discovery - even today some people don't. When the bike falls it leans, and thus to effect a proper turn the bike does lean.
The issue with leaning is that at some point the tires will lift off the ground - GP riders hang off their bikes so that they can reduce the amount of lean on the bike by changing the center of gravity, the less the bike leans the more traction it has and thus the faster it can go.
The gyroscopic effect, which holds the bike up and makes it stable, is stronger at higher speeds. A big part of learning how to ride is learning to let the bike do what it was designed to do. This can be quite a challenge because the SRs get in the way.
There is a great book on this by Keith Code, actually he's written two, _Twist of the Wrist I and II_. He has also made a video of the same name. Keith runs the Superbike school in SoCal and is the premier motorcycle riding authority. The people I know that really know how to ride have read these books and swear by them. My riding skills have increased by magnitudes since I started reading and applying Keith's technology.