It's a frustrating fact of internet life that new websites can take up to three months to get indexed by the search engines. And that can often result in having your old website still show up in search results.
But another reason your old website could still be showing up is that browsers have memories. That is, they store the last version of a url you've visited in their cached history of your computer usage, and will present it to you again when you type in that domain.
Here's what you can do to get your browser to find your new website:
1. Refresh your browser
Somewhere at the top of the screen next to or inside the bar where you can type in keywords or a full url, is usually a circular arrow. That is your refresh button. Click on that to tell your browser to find the newest version of the url you are looking at.
2. Clear your browser history and cache
It's a good idea to clear these functions of your browser every week anyway, to keep your computer running in top form. Be sure to clear the history and cache completely. I cleared my a couple days ago and already have nearly 4000 sites in my browser history. They build up fast because every time you check in on Facebook and go to different pages there, your browser reads that as a separate website to add to its collection.
For example, in the Chrome browser there is an option to clear back to "the beginning of time" -- meaning, to when the computer was first used. If you have cleared your cache from the last week or month, you might not be clearing back far enough. This is usually the root problem when your designer is seeing the new site with no trouble, but you are still seeing the old site.
3. Give the search engines more information
Usually when a website is updated or redesigned, new pages are added. For my clients, those pages are usually titled things like depression, anxiety, couples, bulimia, past lives, hydrotherapy, reiki, and so on.
So instead of typing in the url for your homepage -- such as YourDomainName.com -- try searching for just your own name and the word counseling or therapy (which ever is used in your website header), and the word depression -- Jones counseling depression. Make them all separate words and don't include the dot com.
Usually then the search results will include your niche specific page for depression (or whatever page title keyword you used). Click on that search result link that come up to help train the search engine to find the new website.
It takes a while for new websites to get indexed by the search engines. Some are faster than others. It's a frequency of use issue. The more you search for your name plus counseling /therapy plus depression, the more the search engines get used to looking for pages with that combination of words, and the faster they will learn to find your new site.
While following step #3 won't increase your page rank or do anything else for your basic search engine optimization, searching in the way I've described can help speed the process of recognizing and returning your new site in search results.