How do I get bookkeeping experience without a bookkeeping job?
If you have done a bookkeeping course that is the first step. Next, you have to become a bit creative about your resume as landing your first job is the hardest to get.
Volunteer in a place that needs someone to look at the books.
In any job you are doing tell them you have bookkeeping qualifications and ask if there is any chance for bookkeeping work.
Help a not-for-profit organization. Small not-for-profits often have difficulties finding willing competent treasurers.