If you want to delete your Instagram account then it is only possible via website, not the mobile app. Follow these steps.
1) Access the delete Instagram account page. Now, log in to your Instagram account from browser.
2) Next to “Why are you deleting your account,” Select the reason why you want to delete your Instagram account from the drop-down list.
3) Enter your password and then click on “Permanently delete my account” button.
You have successfully deleted your Instagram account once in for all.
Source: How to Delete Your Instagram Account?
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