I'm sorry you're having trouble publishing your site. Usually it isn't necessary to delete the default under construction page. As long as your website's home page is named index.htm, index.html, or index.php it will show up before the default page (named default.htm) does. This is due to Directory Indexing in your account, for more information on that, please see our article What Directory should I Upload my Files to.
While checking into this for you, I do not see any site files in your public_html directory (your site's document root). How are you creating your site and did you receive any errors when you published? Please feel free to update your question or add a comment below letting us know how you are building your site and the steps to take to recreate any errors you are seeing.
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