The answer depends on quite many parameters, such as:
- how long are you running the project?
- how flexible the client is?
- what is the company's management approach?
- how good is the collaboration with the developers?
The scenario you described used to occur at small companies, lack or poor management, trying to deliver as many projects as soon as possible, without concerning about the quality.
What you can do, based on my opinion, in short term is to document every risk you see. You test the website in one day, do your best, but document every impediment you find (email, jira, whichever tracking tool) and let the client know about the risk. It will be their responsibility if they don't postpone the release. Your job is to find the critical points of the application.
On long term you should try to convince the client/management to be a little bit more organized, time box the testing too, not only the development. I would suggest to use scrum because it allows many releases and it's more structured, however I don't think agile is applicable for every software development case.