Is the website designing business over-saturated?
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It depends on what kind of website design company you want to build. There are many companies who offer template-based sites and these types of businesses quickly find that they slip into commodity-based pricing structures that result in only being able to win business if you offer the most benefits for the least amount of money. This might work in some industries and indeed there are some web design firms that have made this the only way they do business--and some are successful at it--extremely successful.
Having said that, I would say that those successes are the exception and not the norm. The best way to remain competitive in the web design space is to increase your skill set to include some of the new web 2.0 technologies like building blogs and online communities or, at the very least, designing the front-end or user interface for these types of sites.
You can also add more to your value proposition than just design. Do you understand internet marketing or usability? If so, mention those selling points when talking with potential clients and you'll see your value to them increase.
Also, quality service is always a great differentiator. Believe it or not, many companies proclaim to be focused on providing top-notch customer service, but really are not customer-focused at all.
Make it happen,