I have recently got a new job with the county I live in but I am 16 weeks pregnant. I know they offer paid leave for employees that have been working for one year but I will need leave before then. Will I lose my job or are there other options for me?

You will need to discuss this with your HR department. Every company is different.

You might qualify under FMLA to keep your job and be given unpaid leave.

You will not get paid leave of course, but may have some sick time or vacation time they allow you to use.

I will assume they were aware of your pregnancy when you were hired, was this not discussed then ?

But you need to discuss it with HR to determine what they will do.