When you are self-employed (which includes being an independent contractor) then you are responsible for making all the tax payments you owe for the income you are making from the self-employment since no one withholds that tax for you. This includes not just income tax, but also your Social Security and Medicare taxes (together known as FICA taxes). A lot of self-employed people when they first start out overlook the FICA taxes because as an employee they never had to deal with those. The problem with FICA tax is that, unlike income tax, there are no non-business deductions that reduce it. So even if you can reduce your income tax on the business quite a bit with personal deductions, it does not help with the FICA taxes at all. This can lead to much higher tax bills than the taxpayer expects.
Some others here have suggested you look at all the business deductions you could take to reduce the business taxable income, which of course is a good thing to do since it will reduce both your income tax and FICA tax. But since you used turbo tax you should have already taken most of those into account as that program does a pretty good job of prompting you for the various deductions, etc., that you may be eligible to take, especially if you used the more robust version of the program for sole proprietors. So I assume you did that already.
Getting the installment agreement is good first step in resolving the balance due for this year. Getting an installment agreement now cuts your late payment penalty in half and ensures you won’t have to deal with ongoing collection efforts by the IRS, which get increasingly more agressive the longer the tax debt goes unpaid and ignored by the taxpayer. For this year and future years you need to start setting aside the money to pay the tax and remit them in estimated payments to the IRS. Form 1040-ES is used to do that. The instructions for the form explains how it works and has a worksheet to help you get the estimates right. Turbo tax can do the estimates for you, too, and print out the payment coupons you need to use for the estimated payments.
You are correct that the IRS does not often simply forgive (what the tax law refers to as “compromise”) a tax debt. It does have an offer-in-compromise (OIC) program in which it will sometimes settle a tax debt for less than the full amount owed based on the taxpayer’s inability to pay. But if you are able to fully pay the debt through an installment agreement you would not be eligible for that relief. The basic rule of thumb for an OIC is that the taxpayer must offer more money than the IRS could ever possibly collect from the taxpayer before the statute of limitations to collect runs out for the IRS to accept the offer. This usually means having relatives, friends, or some other outside sources that are willing to help you fund the offer.

