No, you won't be taxed twice. Your W-2 shows employer's contribution to your HSA account, and the Form 1099-SA reports the distribution form the account. HSA distributions taken to pay for eligible medical expenses are not taxable, but still must be reported to the IRS.
Your W-2 shows your contributions in box 12, this means contributions via payroll deduction and includes money from your payroll contributions and any employer match. TurboTax will record this as taxable until you answer the questions to indicate that you had a qualifying HDHP and no other disqualifying insurance.
The 1099 form reports of distributions or withdrawals from the HSA. These will be considered taxable until you answer the question that says you only spent the money for qualified medical expenses.
Both contributions and withdrawals must be reported to the IRS on form 8889, and what you report must match with the HSA bank sends to the IRS, otherwise, they will be sending you a letter and may assess additional tax due to a misunderstanding.