It is unbelievable that the myth of the weight lifting stunt growth still exists. The answer is no, weight lifting is not going to stunt your growth.
First of all it is very commendable that you are spending your time in the gym. Most of young people these days don’t even know how the gym looks like and not to mention something else.
Also, I presume that you have personal trainer because you are young and you needed someone to fill you in with the basics. So instead of being shy you should ask him your question and you would know the answer right away.
Also you must be aware of the fact that boys need more time to develop than girls.