I'm originally from the UK and there is (or at least used to be) something called the "Cycling proficiency certificate" or something like this. Many schools or youth clubs offered these and they were supported by the police and traffic departments. I suggest you find something similar as an evening class, and this may help to convince your mother that you are least aware of the road rules and have some basic bike handling competency.
[EDIT: it's now called "Bikeability":
"The Cycling Proficiency Test was a test given by Royal Society for the Prevention of Accidents (RoSPA) which served as a minimum recommended standard for cycling on British roads. It has been superseded by the new National Standards for Cycle Training, branded Bikeability, in England.[1] "]
From my own experience as both a kid (I stopped riding when I was about 16) and having returned to cycling this year, here are a few recommendations:
1) Ride within your limits and confidence - don't be embarrassed to get off the road if you are concerned for your safety. The law in the UK regarding riding on pavements (pedestrian sidewalks) is somewhat discretionary - as long as you give way to pedestrians and ride at a safe speed, you are unlikely to be fined, particularly if you voice safety concerns.
2) Wear reflective clothing - yellow/orange flouro vest / belt etc. - this really makes a difference to your visibility to drivers, especially at night. Stick reflectors on everything - wheels (often they have them), pedals, and even your ankles (moving reflectors are easy for drivers to notice)
3) Use your lights in daytime - normally flashing. A second flashing front white light in addition to a steady one is a good combination at night.
4) DON'T wear headphones - you need all of your senses when riding a bike.
5) Wear a helmet - not law in the UK
AFAIK, but highly recommended if riding in traffic (and anywhere else IMHO)
6) Ride defensively - assume that cars will not have seen you, so you need to make eye contact if you are going to do anything risky. Always scan the road, cars and driveways ahead of you to look for potential threats. A rear-view mirror is really useful for knowing how close traffic is behind you.
7) Avoid riding close to parked cars - sooner or later someone will open their car door without looking. Leave a meter between you and the car unless you see it's unoccupied. Ride in a straight line so that cars can assess your speed and direction.
8) Learn the road rules and obey them - don't jump red lights - it's very easy to mis-judge a car's speed and direction, and you might have a mechanical. problem half way across a junction.
I'm sure there a few more, but this should get you started.
Depending on your location there may be options to use cycle paths, parks or quieter roads. Study route options to avoid busy roads.
Cycling is relatively safe, and whilst you can have bad luck and be hit by an inattentive driver, you can reduce the risk significantly by learning to predict potentially dangerous situations, e.g. cars coming out of driveways but looking the other direction, dog walkers, children playing with balls, buses pulling out etc....
Good luck!
BTW, check out some the Global Cycling Network videos on YouTube - they are based in the UK, and have several videos on various aspects of safety.