The Radeon X300 is a basic graphics card with only 128Mb RAM but even
so it should be up to You Tube video and iPlayer ! It also shouldn't
be a problem with 1Gb RAM though as ever the more you have the better.

What can cause problems with both You Tube and iPlayer is a generally
poor internet connection so what download speed are you getting ?
Check this with www.speedtest.net using a selection of servers and see
what your overall performance is like. Anything less than 3Mb is going
to cause problems with iPlayer and less than 2Mb is going to make You
Tube pause at times. You can also do a Ping Test which measures packet
loss and jitter ( basically a line quality check ) but that needs Java
installed to work.