To clarify, did the fish break the tippet, or did it come free from the hook? Those are two different issues, with two different solutions.
If it’s the former, can you successfully fish a heavier tippet? You may not be able to if using a small fly for a delicate presentation, but if you’re fishing a bigger dry fly or a big nymph you can use surprisingly heavy tippet without reducing your effectiveness IMO…Certainly 4x, and often 3x. Those should be able to handle a 20” Trout in most scenarios, even if you’re a little sloppy technique wise. If you’re fishing streamers, you can go heavier than that even. You generally want to use as heavy of tippet as possible to still be effective in getting strikes for a given fishing scenario. Assuming you have the tippet size figured out right, how did you have the drag on your reel set? The lighter the tippet, the lighter you need to keep the drag to avoid the fish from breaking it, thus giving the fish more of an advantage…back to why you want to fish as heavy of tippet as you can get away with. Your reel should be fine in handling the drag on an occasional 20” fish, but recognize that for consistent big fish angling, a disc drag system is generally more effective and reliable than a click/pawl in terms of engaging smoothly and consistently. If you’re suffering breakoffs it’s also important to know how to fight the fish. It’s difficult at times, but try to not let the fish get downstream of you, especially into heavy or fast water…this magnifies the weight of the fish and the strain on your tippet, and/or your reel. Easier said than done with a big fish…that’s part of the fun. If it does, you’re often not going to be able to reel or fight it back up to you…you’ll need to chase it downstream into the next bit of calmer water and try to land it there.
If the fish threw the hook, that’s a bit of a harder problem to solve with any consistency. It happens. To everyone. Make sure you’re getting a good initial hookset, and work to keep the line tight on the fish at all times. One of the hardest scenarios I run into is when I hook a fish upstream of me, and it immediately runs downstream. You often can’t reel fast enough with a 1:1 ratio fly reel (It’s much easier with a spinning reel where you have a 5 or 6:1 gear ratio advantage), and you can’t strip line in fast enough either sometimes to keep the line tight. I know I got a good hookset, but when I catch up to the line, the fish is gone…Got off while the line was slack and the barbless hook wasn’t under tension in its jaw. The other scenario fish get off after a good hookset is usually while landing it. Use a net for big fish, and recognize when a big fish isn’t ready to be landed yet. When it’s time, raise the rod tip and try to keep the fish’s head out of the water if you can as you bring it toward you and into the net. This will limit its ability to make a last second bolt at the net. As others have said, you typically lose a fish either right away in the fight, or at the end when landing it. After a good hookset, it’s pretty rare to lose one mid-fight as long as you keep the line tight, though it does happen on occasion…Try as best you can to avoid letting the fish get into any structure in the water that it can wrap the line around.