To find TV shows to watch online depends on what type of show and the original broadcast date. Many channels have an online catch up or demand websites where you can watch free shows. It may be possible to watch some TV shows via a video streaming website, such a YouTube.
There are many websites which offer free television shows online. Most of the regular networks such as CBS, ABC, NBC, and Fox all have some or most of their regular shows online to watch. Other sites such as Hulu has free television online as well.
There are many places to watch TV online legally. There is Project Free TV which includes many classic movies and shows, YouTube now includes many movies and TV shows and channels and even Hulu has many free shows. These are also usually offered on network websites as well.
Any television equipped with an antenna can allow one to watch free television shoes legally in the US. Also, many of the major broadcast networks allow viewers to watch selected episodes of shows online from their websites.
If you by that mean "cartoon network shows online" then try cartoonnetwork.com (free for paying tv customers). For 90's cartoon network shows you should try funniermoments.com (free), netflix.com (paid), amazon.com (paid), hulu.com (free) and crackle.com (free).
Users can watch television online for free on Amazon, Netflix, and Hulu. They all offer free trials that one can use to watch television for free. Just cancel the subscription when the free trial is over.