If the range is 12-22 and your result is 10 then it's below range!
Is there another GP you can see? If so go back and ask for further testing to check Thyroid antibodies, Vitamin D, folate, B12 & ferritin.
If you can't face the battle then yes it's easy to get private tests. Many of us on here do so
Medichecks Thyroid plus ultra vitamin or Blue Horizon Thyroid plus eleven are the most popular choice. Both companies often have money off offers on.
DIY finger prick test or option to pay extra for private blood draw.
All thyroid tests should be done as early as possible in morning and fasting. This will give the highest TSH, which is as you have discovered all many GP's incorrectly look at.
When you get retest come back with a new post for advice on how to proceed.
Meanwhile read posts on here, & the main site Thyroid UK to learn as much as possible