Hello Sophie, cooking delicate fish fillets can definitely be frustrating when they stick or break apart, but with a few adjustments to your technique, you will be able to keep them completely intact and get a nice golden crust.
First, excess moisture is the most common reason fish sticks and tears. Before cooking, use paper towels to thoroughly pat both sides of the fillets completely dry. If you like, you can lightly dust the fish with a thin layer of flour or cornstarch and shake off the excess, which creates a protective barrier and helps with browning.
Second, make sure both your pan and your cooking oil are properly heated before the fish goes in. Place a skillet over medium-high heat with a couple of tablespoons of neutral cooking oil. Wait until the oil is shimmering hot. If the pan or oil is too cold, the fish will absorb the oil and stick instantly.
Third, do not move the fish too early. Once the fillet is in the pan, leave it undisturbed for about 3 to 4 minutes. As the bottom cooks, it will form a crisp, golden crust that naturally releases from the pan surface. If you try to slide a spatula underneath and feel resistance, give it another minute.
Finally, only flip the fillet once. Use a thin, flexible fish spatula rather than tongs or a thick spatula, gently slide it completely under the fillet to support the whole piece, and flip it over to finish cooking. I hope this helps you make a wonderful dinner next time.