Noone in my immediate family has a thyroid problem either so I guess it can be "normal" if you can call that normal, haha.
Be sure that you get a T4 free, T3 free and thyroid antibodies done as well as the other standard tests most docs do. Your tests should be back within a week, sometimes within just a few days depending on the lab.
If your tests show a thyroid problem then you and your doctor should discuss treatment, this usually means putting you on thyroid meds, if your doc only prescribes synthroid and nothing else you may want a different doc, not everyone does well on synthroid and there are different brands.
Keep in mind if you do go on meds that it generally takes 4 to 6 weeks to notice much of a change in how you feel. Retesting should be done after 6 weeks on meds to see how your body is adjusting to the meds.
Remember, you can always pop on here and ask any questions and be sure to get a copy of your blood tests and if you want post the results along with the lab ranges here on the boards so we can help you understand them, not all docs will explain things very well.