Hi,reto0110!
I looked on your tinkpad specifications. I've found that it has 256M RAM, correct me if you have more.
Such not very big amount means, that you need swap, in order not to catch dead freezes. PLease describe more about what TinyCore version do you use, the way you installed it (or maybe running from live flash) and your HDD partitioning scheme.
But my opinion is that on 256M you will not be able to run full-blown modern browsers like firefox or chromium. And viewing video directly in browser is very CPU and RAM consumpting, so i propose using more ascetic browser (lynx, dillo, fifth) for links choosing, then download video with youtube-dl, and watch it using mplayer.
Regards!