Is this a common practice?
You could say yes, but it depends on your negotiation skills and also on what terms you are leaving the company (if you burned bridges they most probably won't be happy to compensate your extra work). It also depends if it is worth asking for more money, given the amount of extra time you will work, so you don't come across as too greedy.
Experienced professionals could easily recognize such opportunities and negotiate good compensation, turning the "puts us in a tough situation" to their benefit. This ability mostly comes with time, as you face new job experiences (like this one).
What are some benchmarks that can be used to come up with a fair compensation?
Assuming that your new job has a better salary, you could use the difference to increase your current compensation, so your income does not decrease during that time (as mentioned in comments).
It could also be the case that postponing your start date at the other company would represent some expenses on your or the new company's behalf, such as (these are the ones I could think of):
- legal expenses required to postpone contract
- no health coverage or other benefits during that period
- other salary compensations delayed (retirement or lay-off bonuses)
- deductions from your salary by the new company because of other measures they took
Now that you have a better idea of how much you could ask for compensation, I can think of two ways you could negotiate to obtain them:
- Charge them on a weekly basis, dividing that amount between the extra weeks
- Negotiate an upfront payment of the total amount that covers the expenses.
Remember to add any extra compensation you decide should be mentioned when proposing this.
All this being said, it was not clear to me if you already agreed on a price for those 3 weeks. If you have not, it would be worth attempting to get better compensation. If you already accepted an offer, however, it would be unprofessional to go back on your agreement. In that case, let it be a lesson for future salary negotiations of this sort you may encounter. Hope this helps.