It's been 4 months sense I would have gotten pregnant, the first month after unprotected sex I missed my period, the second month I bled around the 4th day for only that morning and the night before I had sex which could have caused the bleeding, later that month ,at the end, I had sex again one night and the morning after ,or possibly right after having sex, I bled for about 3 to 4 days , and now the fourth month I haven't yet started my period but right now I am having brown and white thick discharge no odor. Also this whole time, besides this week, I have been throwing up after everything I eat, my stomach cramps up, and I got headaches from nasty smells and loud noises for about a month or two after the night I might of gotten pregnant. I've taken four pregnancy tests all came back negative and I have not gone to the doctors for the testing of pregnancy but I have gone to the doctor for my throwing up and constant headaches and stomach cramps but all tests were negative, I still don't know what is wrong with me after 4 months of this, I was wondering if I could be pregnant?

It is possible, but other possible issues also
Assume you are testing with first urine of the morning.

But if you think you could be 16 weeks pregnant you are way, way over due for pregnancy test at doctor and to start getting scans