Is it correct?
To be honest, this site disappointed me. I began to give him less time. Thanks for the commentary and your attention, dear friend.

Question

Answer

Don't use "him" because a website has no gender.
There are several ways you can say this:

To be honest, this site disappointed me. I began to spend/spending less time on it.
To be honest, this site disappointed me. I found myself spending less time on it.
 To be honest, this site disappointed me. I started to spend/spending less time on it.

Answer

Hi Roman,
I would say "I began to dedicate less time to it"