I would like to know what should be done when a search engine has  penalized my website for Low Link Popularity

Link popularity is one of the most important factor for driving traffic to a website.
How To Check Your Website Link Popularity?
There are many free online tools that can help you to check your website link popularity. One of the tool is Submit Express Link Popularity Check Tool.
Search engines do not trust on new & unpopular websites. For every single keyword you enter in search engines, they evaluate millions of web pages in order to show you the best result. You should take note of the following points for not being penalized.
1 – Do Not Manual Submission To Search Engines
If you are using manual submission to index your sites in search engines, it might take a long time to see the result. In order for search engines to index your new websites, simply add a link on other popular websites pointing back to your website. The whole process will take only few days.
2 – Your Websites Can Be Delisted From Search Engines
If you do not earn enough link popularity search engines can delisted your website. One of the reasons is that the contents of your page or the links pointing to your page changed significantly.
3 – Search Engines Loves Higher Link Popularity Websites
Search engines depend on link popularity to rank websites in the search engine results. From search engines point of view, a link from site X to site Y can be count as a vote given by site X to site Y. The more votes your site has, the higher search engine rankings will be given.
How Many Links Is Enough?
It depends. Every website need a basic amount of links to maintain its ranking in search engines. If competition is higher, then you need more incoming links to maintain your rankings. If your competition is less, then you need less links to have high search engine rankings.

typically, search engines wont penalize you for not having links. They only penalize those that implore black hat techniques.

If you have a low level of links, obviously the goal would be to gain more backlinks from RELEVANT sites that are not Black Hatters.