Suppose that I provide support for a much older relative. I worry that in the unlikely case if I die before the relative, the relative will not be provided for. Is there a life insurance that pays only if I die before the relative? Since I am much younger and healthier than the relative, such a product should be much cheaper than the term life insurance. Does it exist?
I don't think the product you're seeking exists. However, there may be a way to simulate it. As keshlam points out in another answer, you are basically looking for a policy that would payout enough to purchase an annuity for your relative. Note that the price of that annuity should decrease every year, so even if you could purchase an open-ended term, you would ideally want one that decreases the payout every year to coincide with the cost of the annuity. The fact that you may need to purchase a term policy that is too long, just in case, can be somewhat balanced out by the fact that you need less money to cover the annuity the farther out you go in time. This leads to "laddering".
Last year I was looking for a similar policy (for a different reason) that would taper down the payout each year, but after talking to a couple of insurance brokers it doesn't seem to exist (at least the two brokers hadn't heard of it in the US). I ended up doing a mini version of it myself by laddering two term policies together. In theory you could ladder as many policies together as you wish. For example, if the cost of the annuity today is $600K for a 75 year old, it might only be $400K for an 82 year old and $200K for a 90 year old. So you could purchase 3 policies of $200K each, one for each of 5, 10, and 15 years. This would obviously be less expensive than a single $600K policy for 15 years.
Tip: You can save even more by adding term riders to another term policy instead of purchasing multiple separate policies, as explained here.
A Term Life Insurance. is exactly that. It pays out a benefit in the event of your death in a specific period of time - i.e. a "term".
Term Life policies are cancellable at any time, so when your relative dies you can cancel it without penalty. This is exactly what you are looking for. You buy one for a period of time, cancel it if the relative dies and renew it if they are still alive at the end of the term.
It sounds like what you are trying to do amounts to carrying enough insurance on yourself, with the relative as beneficiary, to permit the relative to buy an annuity which will cover the rest of their life. Ideally you'd want to adjust the insurance payout each year to reflect inflation on one hand and the shortening expected lifespan on the other.
That should be something you can calculate. If you need help doing so, an insurance agent should be able to provide that help; these are both insurance products. I haven't heard of an off-the-shelf service which combines them. But I'm not an insurance agent so that's hardly a surprise.
Of course you might want to leave the decision of whether to actually purchase the annuity up to the beneficiary. Annuities are reassuring since they guarantee a payout, but they do take a processing fee out of the money and most don't adjust for inflation (or at least, not without an even bigger fee to cover that risk). If the recipient already has someone managing their investments, they might do better putting the money there. You need to decide whether you want to take that decision on their behalf or not.