At the beginning of your debt-free process, the debt snowball method is quicker. You will see that you have paid off smaller debts in as little as 12-24 months. Once you have paid off the smallest debts though and start getting into the bigger ones with high interest, you might find that your debt elimination stalls a bit.
Using the debt snowball method though will get rid of unwanted small debt much faster than the debt avalanche method.