A great solution to this kind of a problem is by first sitting down with the designer, present your budget, and work things out from there. In one way or another, you will get something close or greater.
Client tends to rush a project and the result is a not so well done website. It will result to a bad portfolio for the client as well as designer. In such a case, the blame is on the client and it is the work of the designer to sit down with the client and formulate a working schedule.
Nothing in this world will ever be perfect. That is one thing a client must understand and accept. Delay in launching a website occurs because the client always tries to change small information in a website. It can even be a comma in a sentence or moving a picture 1 pixel more to a side. In such a case, the designer should make it clear that they can only work with what they first agreed and any extra change is charged.