Some websites I am accessing through Tor recognise that I am masking my IP address. They are blocking my access requiring that I access the website via my actual IP address from my service provider, of course this defeats the purpose of using Tor. How do I bypass this?
There are public lists of Tor exit nodes that site operators can use to identify Tor users. Unfortunately, using Tor necessitates the use of an exit node, which means that it will be possible to identify Tor users.
A possible solution without Tor is to use a Virtual Private Network (VPN) to mask your original IP address. VPNs are subject to being treated in the same way as Tor exit nodes but they are much more plentiful and it is difficult to maintain an accurate list of VPNs.
If you want Tor-level anonymity, simply using VPN services is in no way adequate. For at least some websites that block Tor exits, it may be sufficient to use "anonymous" HTTP/HTTPS proxies. HideMyAss has a good list of free proxies that are probably not botnet slaves. There are browser add-ons to manage proxies.
If proxies aren't enough, you can tunnel TCP-mode VPNs through Tor. The simplest approach uses Whonix with a VPN client in the workstation VM. To best maintain Tor-level anonymity, it's easiest to use free VPNs, such as SecurityKISS or VPN Gate. You must set everything up using Tor, to protect anonymity. However, free VPNs often have bandwidth and/or usage caps.
In order to use paid VPNs, you must make all arrangements via Tor, and pay as anonymously as possible. Cash through the mail is a good option, but slow. It's faster to use Bitcoins. But you must anonymize them by multiple transfers (using Tor) through mixing services (such as Bitcoin Fog) among multiple Multibit wallets in Whonix VMs.