"Modern fighter jet" and "F-35" are not synonymous. The F-35 is just one out of 20-30 fighter jets currently in full-scale production (depending on how you define "full-scale"), and it's not representative of the rest of them at all.
The majority of fighter jets have their skin made out of aluminum, of about the same thickness or a little thinner than on passenger airliners. Except in the wings, it's lightly loaded and not structurally essential, so it doesn't matter that aluminum's strength falls greatly at elevated temperatures.
There is also a lot less force put on the airframe at high speeds than they're designed to handle. Fighters are only able to maneuver well at 200-400 knots, well below the speed of sound. On the average, they travel at Mach 0.7-0.8, similar to passenger jets or slower, as they're not as aerodynamically clean. Supersonic flight is only used for interception, in a straight line, for short periods of time, to get at another aircraft.
The F-35's skin is made out of CYCOM 977 epoxy resin in most fuselage sections, CYCOM 5250 bismaleimide resin in the wings and some hotter surfaces, reinforced by ~700 ksi tensile, intermediate modulus carbon fiber, mostly Hexcel IM7. Bismaleimides generally offer better heat resistance than aluminum; epoxies are comparable to Al, depending on the specific alloys and resins being compared.
That said, the F-35 is about the slowest currently produced non-training fighter, and aluminum has proven to handle the higher heat stress in faster fighters just fine. But CFRP is a bit lighter, less prone to permanent bending than metal and provides much better adhesion with the outer layer of resin-embedded RAM. The latter two properties make composites essential for VLO (all-aspect stealth) aircraft.
P.S. More on various resins: What are typical resins used in carbon fiber for aircraft? Are any resins impossible to recycle?