Is it worth buying a lighter bike? If so what kind?
IMHO, it's not the weight that is hurting you the most. While weight makes a difference and a lighter bike would be much better, it's too common in cycling world for people to use weight as a proxy quality and performance. Any twit with scales can measure it.  That said, there is no doubt more suitable bike would make longer trips faster, and a better quality bike even more so, but only you can decide if the cost is worth it.
Weight only affects hills and acceleration. For a long ride mostly on the road we can practically ignore the effects of acceleration.
On the hills, lets say you shaved 10kg off the bike, your total weight is now 75kg instead of 85kg. From this calculator, a 100M climb over 10km will take 38 minutes on the lighter bike vs 40 minutes on the heavier bike. Using these rough numbers, the calculator comes up with figures that are close enough to help you decide &quot;Is it worth it&quot;
However, resistance and power loss from a bike such as this are a big factor that cannot be ignored.  New tires that are a hybrid might be more suitable—a bit of lost time in the forest for faster riding on the rest of the trip might be a trade off. Something like Schwalbe Hurricane would be a goods starting point.  The shock and fork should be tightened as much as possible, since I doubt they have lockout.  Your position on the bike, getting more aerodynamic would make a big difference—bar ends could help with this.
While the perfect bike for you sounds like a gravel bike, there is no need to head down the path of very expensive to get a significant improvement in performance. A hard tail 29&quot; MTB running something like the Hurricanes would be ideal and handle the forest section with ease. A drop bar bike running 35mm tires would be good if you are a skilled rider and happy to take the wins on the road for some drawbacks in forest sections.
Before buying, ask around and see if someone you know has a bike they may be prepared to lend you for a couple of rides—you might be surprised who has an old bike in need of an outing...