The flyback diode slows down the rate at which the current reduces in the relay coil. That can slow down the rate at which the contacts open, which isn't a good idea with large loads, as it can lead to contact burning.
There are a few alternatives to a flyback diode, but all will increase the maximum voltage that the switching device feeding the relay is subjected to. However, with each alternative, the maximum voltage is quite well defined and is far less than you would get without any flyback diode.
The simplest alternative is a resistor in parallel with the relay coil. If you have a resistor whose value is 10 times the relay coil resistance, the maximum voltage that the switching device will be subjected to will be about 10 times the supply voltage, so about 120 V for a 12 V supply. The coil current will reduce about 10 times as fast with the resistor in place as it would with a flyback diode.
A resistor like that will take about 1/10th of the power of the coil, in addition to the coil power. If that matters, you can put the flyback diode in series with it.
The 10 times factor is not fixed. The larger it is, the larger the maximum voltage and the faster the relay turns off.
Another alternative to a flyback diode is a TVS or transient voltage suppressor. Here is an example:-
https://uk.farnell.com/taiwan-semiconductor/sa36ca-r0/tvs-diode-500w-36v-bidir-do-204ac/dp/2677629
That will limit the voltage to about 36 V so that the coil current will still fall about 3 times faster than it would with a flyback diode. The voltage suppressor won't take any current during normal running. You need to have a bidirectional TVS.