I had the same problem for years, and have opted to buy new bins last year.
There are a lot of bins sporting extra removable buckets in them (has something to do with being able to clean them better).
When the time arrives to clean my bins, I just take the removable bucket outside, tie off my trash bag, throw it away and take the removable bucket back in the house.
An old study-partner of mine always put a real bucket on the bottom of his trashcan, because he always punctured his trash bag at the bottom so it was easier to remove from the trashcan.
The bucket would contain the liquids and he only had to clean that instead of the whole trashcan!
lifehacks.stackexchange.com/users/128/RichardBernards
Use two bags per load.
When the bin is full, tie off the top tightly, so that nothing can leak out, and then open another bag over the top, dumping the first bag into the second.
Tie it off, and carry it off.
The first bag should be upside down inside the second.
Pro:
You use up the nasty bags twice as fast! :P
lifehacks.stackexchange.com/users/25/J. Musser
Cover all of the holes with a few big pieces of plastic wrap or aluminum foil.
Use duct tape to secure it.
Alternative Solution Before placing any trash in the trash bag, place a wad of newspaper at the bottom of the bag.
This will help to absorb any juices that are in the bag.
lifehacks.stackexchange.com/users/20/michaelpri
Just tell me, you can predict what could be the cause of the leak?
Then you can just put all the possible leakers (I don't think you would have really many of them) in additinal small cellophane bags.
I, for example, have a big roll of those bags and sometimes use them in the case something I bin would leak.
lifehacks.stackexchange.com/users/21/nicael
Double up bags
(more wasteful) Cover each hole with an "X" of duct tape after unfolding the bag and before lining the can.
lifehacks.stackexchange.com/users/15/Mooseman
Instead of walking about with a bin bag, take the indoor bin with you when you need to empty it.
Then, once outside, transfer the bag from the indoor to the outdoor bin in one smooth steady motion.
By limiting the amount of time the bin bag is jostling about, you should reduce the amount of sloshing of the liquid within it.
Another advantage is that you're now outside to tip away any bin juice that may be in the bottom of the bin too.
lifehacks.stackexchange.com/users/5099/Easy Tiger
The bin bags (trash bags) I have just bought in a roll of 100 of have small holes in the bottom, presumably so that they can compress when tied off. The downside of this is that if the bin contains something liquid, or something that has become liquid, it leaks out of the bottom when I take the bin bag out of the bin.  Is there a way of avoiding this very unpleasant situation?
How do I get the bin bag (trash bag) outside without it dripping all over me?
lifehacks.stackexchange.com
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lifehacks.stackexchange.com/users/160/fredley
One simple way to avoid this problem is to simply take the entire bin outside when taking the rubbish out. In fact, some bins will have a removable bucket to allow you to do this. However, another simply method involves simply doubling up on bin bags, placing the original bin bag upside down in another bin bag. A slightly less simple method would also be to cover the holes in the bottom of the bag using either plastic wrap or aluminium. The options are to get a bin with a removable bucket, open another bag over the full bag and tie up both, cover the holes with foil or wrap or place newspaper at the bottom as a sponge, put leaking items into separate plastic bags or just carry the whole bin outside to empty it.
Get a bin with an extra removable bucket in it and take the bags out in the bucket. When the bag is full tie it and then open another bag over the top of it, tie that bag too and then carry out. Cover the holes with plastic wrap or aluminium foil or put some newspaper at the bottom to soak up any liquids. Put any leaking items into separate plastic bags and then into the bin. Double up the bags or cover the holes with duct tape. Take the whole bin outside and empty it rather than carrying the bag out.