Doing a 2-Way, 2-Gang Light Switch
The old switch
Strictly speaking, you need to find out what those three wires are connected to. If I had to guess, though, I'd feel quite confident that the wires are as labeled here:

The bridging wire (called a &quot;jumper&quot; in my world) tells the tale. The power supply is the only wire that would be connected to both switches, and thus to two terminals, and thus need a jumper.
The previous sparkmonkey's choice of terminals gives one pause, but the circuit works. The gauge of the jumper is too small; it should match the load wires. And you should check that your local code allows two wires to be crushed into one terminal.
The New Switch
You should consult the documentation that came with the switch; there should be a diagram showing the purpose of each terminal. If I had to guess, though, I'd feel quite confident that the terminals are apportioned to the switches like this:

L1 is switched between L11 and L12. L2 is switched between L21 and L22.
You must connect the power wire to L1 and L2.
If you are allowed to connect two wires to one terminal, then use one jumper. Make the jumper from wire of the same gauge as the load wires. Connect the power wire and one end of the jumper to L1, and the other end of the jumper to L2.
If your electrical code forbids two wires on one terminal, then use two jumpers and a wire connector (ie. wire-nut, Marrette, Wago, etc). Connect one jumper to L1, and the other jumper to L2, then use the wire-nut to connect the free ends of the jumpers to the power wire.
In any case, connect Load 1 wire to L12 and Load 2 wire to L22.
If this does not work, the one of my guesses is wrong, and you have some diagnostic work to do. In that case post a follow-up comment and I (or someone here) will guide you through the testing process.