Ask HN: Best practices for giving clients total cost estimates

So a potential new client plops a big involved website project down in my lap and asks "how much?".

I can give them my hourly rate, and I can make a wild guess at approximately how many hours it might take... provided everything goes perfectly according to plan.

But I really struggle with giving clients with budgets a proper estimate.

I find that I can't provide an accurate estimate until I have completed at least 10 or 20% of the work to be done.

But brand new clients are of course a bit hesitant about allowing me to start the clock running without giving them a full estimate.

I also feel extremely pressured by my own estimates once I provide them.

I say 50 hours at my rate, they say OK, and then 25 hours into the project I realize that it's actually going to take me closer to 70 hours.

Is it fair for me to work 20 hours extra for free essentially? Should I be punished for not making an exact estimate of a fairly complicated and involved process?

I am a competent programmer, but I really struggle with the concept of estimates.

Any and all advice you can provide will be appreciated!