First erase the current update-alternatives setup for gcc and g++:

sudo update-alternatives --remove-all gcc 
sudo update-alternatives --remove-all g++

Install Packages

It seems that both gcc-4.3 and gcc-4.4 are installed after install build-essential. However, we can explicitly install the following packages:

sudo apt-get install gcc-4.3 gcc-4.4 g++-4.3 g++-4.4

Install Alternatives

Symbolic links cc and c++ are installed by default. We will install symbol links for gcc and g++, then link cc and c++ to gcc and g++ respectively. (Note that the 10, 20 and 30 options are the priorities for each alternative, where a bigger number is a higher priority.)

sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-4.3 10
sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-4.4 20

sudo update-alternatives --install /usr/bin/g++ g++ /usr/bin/g++-4.3 10
sudo update-alternatives --install /usr/bin/g++ g++ /usr/bin/g++-4.4 20

sudo update-alternatives --install /usr/bin/cc cc /usr/bin/gcc 30
sudo update-alternatives --set cc /usr/bin/gcc

sudo update-alternatives --install /usr/bin/c++ c++ /usr/bin/g++ 30
sudo update-alternatives --set c++ /usr/bin/g++

Configure Alternatives

The last step is configuring the default commands for gcc, g++. It's easy to switch between 4.3 and 4.4 interactively:

sudo update-alternatives --config gcc
sudo update-alternatives --config g++

Or switch using script:

#!/bin/sh

if [ -z "$1" ]; then
    echo "usage: $0 version" 1>&2
    exit 1
fi

if [ ! -f "/usr/bin/gcc-1" ]; then
    echo "no such version gcc/g++ installed" 1>&2
    exit 1
fi

update-alternatives --set gcc "/usr/bin/gcc-$1"
update-alternatives --set g++ "/usr/bin/g++-$1"

It's possible you arrived because something blew up with nvidia. (I'm on cudatoolkit 11.4 and actually downgrading got me out of trouble - your mileage may vary.)

sudo apt install gcc-9 g++-9
sudo mkdir /usr/local/cuda/bin
sudo ln -s /usr/bin/gcc-9 /usr/local/cuda/bin/gcc

WARNING - I recommend using timeshift to make a backup of your working system. https://github.com/teejee2008/timeshift

This will get you the latest gcc 11

sudo add-apt-repository 'deb http://mirrors.kernel.org/ubuntu hirsute main universe'
sudo apt-get install g++-11
sudo update-alternatives --install /usr/bin/g++ g++ /usr/bin/g++-11 100
sudo update-alternatives --install /usr/bin/g++ g++ /usr/bin/g++-11 50
sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-11 100
sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-11 50
sudo update-alternatives --install /usr/bin/cpp cpp-bin /usr/bin/cpp-11 100
sudo update-alternatives --install /usr/bin/cpp cpp-bin /usr/bin/cpp-11 50
sudo update-alternatives --set g++ /usr/bin/g++-11
sudo update-alternatives --set gcc /usr/bin/gcc-11
sudo update-alternatives --set cpp-bin /usr/bin/cpp-11
gcc --version
gcc (Ubuntu 11-20210417-1ubuntu1) 11.0.1 20210417 

Copyright (C) 2021 Free Software Foundation, Inc.

As @Tommy suggested, you should use update-alternatives.
It assigns values to every software of a family, so that it defines the order in which the applications will be called.

It is used to maintain different versions of the same software on a system. In your case, you will be able to use several declinations of gcc, and one will be favoured.

To figure out the current priorities of gcc, type in the command pointed out by @tripleee's comment:

update-alternatives --query gcc

Now, note the priority attributed to gcc-4.4 because you'll need to give a higher one to gcc-3.3.
To set your alternatives, you should have something like this (assuming your gcc installation is located at /usr/bin/gcc-3.3, and gcc-4.4's priority is less than 50):

update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-3.3 50

--edit--

Finally, you can also use the interactive interface of update-alternatives to easily switch between versions. Type update-alternatives --config gcc to be asked to choose the gcc version you want to use among those installed.

--edit 2 --

Now, to fix the CXX environment variable systemwide, you need to put the line indicated by @DipSwitch's in your .bashrc file (this will apply the change only for your user, which is safer in my opinion):

echo 'export CXX=/usr/bin/gcc-3.3' >> ~/.bashrc

gcc-12 is not available in ubuntu 20.04, so we need to compile it from source code, here are the steps which I borrowed from this video:

• Step 1: clone gcc source code and checkout gcc-12 branch

$ git clone https://gcc.gnu.org/git/gcc.git gcc-source
$ cd gcc-source/
$ git branch -a
$ git checkout remotes/origin/releases/gcc-12

• Step 2: make another build dir

Note this is important as running ./configure from within the source directory is not supported as documented here.

$ mkdir ../gcc-12-build
$ cd ../gcc-12-build/
$ ./../gcc-source/configure --prefix=$HOME/install/gcc-12 --enable-languages=c,c++

• Step 3: installing GCC prequisites and run configure again

The missing libraries will be shown in above ./confgiure output, search and install them one by one.

$ apt-cache search MPFR
$ sudo apt-get install libmpfrc++-dev
$ apt-cache search MPC | grep dev
$ sudo apt-get install libmpc-dev
$ apt-cache search GMP | grep dev
$ sudo apt-get install libgmp-dev
$ sudo apt-get install gcc-multilib
$ ./../gcc-source/configure --prefix=$HOME/install/gcc-12 --enable-languages=c,c++

An alternative is to run the download_prerequisites script.

$ cd ../
$ cd gcc-source/
$ ./contrib/download_prerequisites
$ ./../gcc-source/configure --prefix=$HOME/install/gcc-12 --enable-languages=c,c++

• Step 4: compile gcc-12

$ make -j16

Still flex is missing:

$ sudo apt-get install flex
$ ./../gcc-source/configure --prefix=$HOME/install/gcc-12 --enable-languages=c,c++
$ make -j16
$ make install

Another way is to use Ubuntu 22.04 where gcc-12 is available. In Ubuntu 22.04, gcc-12 can be installed with apt:

$ sudo apt install gcc-12