So my website just got indexed by Google, which I am really excited about, but there is a small problem. When you search for the name of my site in google e.g. “mynewwebsite”, my page comes up first in Google rankings, but when you search for it with spaces in between the words, e.g. “my new website” it is not on the first page and I don’t know where it is. Can someone please help me on how to optimize my site so that when someone searches “my new website” in the google search bar it will have a higher ranking? Thank you so much.
Depends on competition and also most important is quality backlinks… Dofollow PR Backlinks works great…
If you want Google to index your blog also by key words, separating the words from the name, use as many SEO instruments as possible. I recommand you first registering the blog in directories, then use the keywords from your name to articles.
You are right for Time is always a factor in SEO. I could say that SEOs are patient because they must be. Patience in waiting for the site results.
It’s all about keywords ruling. Google show differenet results in every keywords.
Illustration:
Keyword is totally different to keywords, so when you search it in Google you will receive different results.
Another is “Yourkeyword” is totally different to “your keyword”
That means, anything changed, add, or omit in your keyword will automatically change the Google Search results.
Time is also a factor when it comes to ranking high in SERP, as your site grows older your rankings will gradually improve.
The key words are very important for site ranking. My opinion is to use the software that submit sites to the different site engines. You can promote your site too in the same way and can earn money.
but that’s not relevant to this topic to discuss here.
<snip/>
Yes, i have domain with “-” and i think google detect that is space. and i get number one for that kayword.
Could you explain how to make backlinks from good page rank sites? 
Pretty simple you have to get lots of backlinks to your site with the keyword anchor text “my new website”. Well not simple, it will take a lot of work. But the key will be if you can get relevant sites with good rankings to add your link.
This can become a very long discussion 
First off Google doesn’t rank Keywords (Google Keywords), but regards description, title in combo with actual content important. So if you want your site to be found on the different words within your domain name, use them in those three!
I thought that with “-” you wont get anything but now I have something more to think about… I`ll try it on few domains to see what is going to happen 
You can try to buy a domain with the domain name as “my-new-website”. I think Google recognizes “-” as space. Hope it helps.
You need to apply some on page and off page task. Include your 2 target keywords mynewwebsite and my new website on your list for the submission. This will also help to improve and build more links. Have more patient.
For getting quick results you use these keyword in title, meta and in content then you will make backlinks by using this keyword from good page rank sites. You will see the difference.
Your website is quite new as you said, your website title is a word without a meaning, but when you separate them might come to some really competitive keywords. You could wait for years to rank high for those terms, or weeks, depends on the keywords and the work you’ll do for this site.
That’s because the first instance is your domain name which you’d expect to rank well for but on the second search with the gaps in between the words you’ve turned it into keyword phrase and clearly don’t rank well for it.
you have to target that kw when you build some links… the results is totally different with spaces…
If you want to rank your domain name as your target keyword then that is not a big task. Only the thing you need to do is some good effort on SEO. But ask yourself if that word can really bring business to you or not? As the website is new I can say your domain name may not be a good business keyword. It’s just the beginning so I would also recommend you to be patience.