First hint:

The Login Form should correspond to the one provided with CB.
While the big Login Page you see (with the red cross) correspond to the core login component of Joomla (com_login) ; and this one doesn't work in conjunction with CB.

Each time the bridges require a log-in action, the user is redirected to Joomla's core login page by default . But, you can change the page the user is redirected too. You may ask : "I'd like to be redirected to the login component of CB". Ok that's possible with recent version of CB.

So how can I change such link ?
Edit file joomphpbb_engine.php .
Look for the string:
/index.php?option=com_login&Itemid=88888888
And then replace it by:
/index.php?option=com_comprofiler&task=login

That's not everything !
Replace the registration page , core link:
/index.php?option=com_registration&task=register
by the CB's one:
/index.php?option=com_comprofiler&task=registers

Now for the logout link, replace the line:
$target = "/index.php?option=logout&return=" . $scriptRel . "/index.php";
by this one:
$target = "/index.php?option=com_comprofiler&task=logout";
You can also remove the login/logout link from your script's template.

That's all folks.