If it's bitter that can mean one of three things: too much heat, too much time, or too much coffee. The solution is properly ground coffee, but I'll address these factors one by one so it's easy to understand why.
French press is a relatively temp-stable method (due to the surface area of the brewing vessel and heat loss ratio) so that's unlikely. As long as you're starting with 200F/93C - 205F/96C water you're golden. Even 208F/98C is fine as long as you're making a larger carafe.
Also a strong suit of French press. Acceptable ranges are between 3 min and 10 min depending on the recipe, roast, and varietal of coffee. Generally speaking, over-extracting (bitterness) is difficult with properly ground coffee.
Okay so this is where it gets complicated. There's two ways of thinking about "too much coffee" - ratio and surface area. In terms of ratio, we're referring to the ratio of ground coffee to water (typically measured in grams). A good "baseline" brew ratio is 1:15 (1 part coffee to every 15 parts of water). That means if your water is 300ml (which is 300g in weight) you want to have 20g of ground coffee. If you change your ratio, without changing your brew vessel, your ground coffee input would change but your water remains the same (e.g. 1:16 = 18.75g:300ml; 1:12 = 25g:300ml; etc.). This being said, all of your proper ratio work can be undone by your ground coffee's surface area.
Why is this? Quite simply because the more surface area your ground coffee has the quicker, and more completely, hot water can extract the coffee flavor from the ground beans. The finer grind you have, the more surface area of coffee can be exposed to hot water at once. So, even if your ratio, time and temperature is correct, those variables are 'keyed' to act on a much smaller surface area.
French press grind coffee works in tandem with the temperature and time - the grind is so large that the water does not penetrate the core of the particle until the water has cooled enough to keep bitterness at bay. In all likelihood, since you've also described a muddiness to your French press coffee, you're probably doing one of two things wrong: 1) grinding it too small. 2) grinding with a bad, or inappropriate, grinder. You wanna get something with large-grind consistency like a modified Hario Skerton, Lido 2, Baratza Virtuoso, or Kuissential Evengrind (essentially a modified Hario). Even quite nice grinders, like the Baratza Sette 270, can't really get good large grind because they're not built for it.
If you're doing all of the above, then you can clean your cup up in two ways: decanting or pouring it through a paper filter before serving.
Anyway I hope that all helps. What grinder are you currently using and what does your grind look like?