There are few unrelated issues with that. The "Web site" cannot crash otherwise an HTTP server crashes which does not really happen, maybe extremely rarely. What could happen is that your PHP code throws an exception when a HTML page is being generated.
Here is the problem: a site PHP settings should be tuned the way the unhandled exception is never shown during production. Some Web hosting providers violate this rule just because people working with this site are illiterate. At least, I faced with such situation once. This is intolerable failure; if this is the case, you should either make your provider to do it right or leave the provider as soon as possible. Ideally, you should have two separate sites: one for development and internal testing, another for production. The internal development side can use both options: to hide all exception handling if you have them unhandled in your code, or show them for debugging purpose. The production server should always hide unhandled PHP exception.
On the development side, you should not let unhandled exception. You need to handle all exceptions on the very top of the stack of your code. As to catching and handling exceptions anywhere in between, you should keep catching and handling exceptions to the absolute minimum. One of such exclusions is: sometimes you need to catch low-level and general exception, use exception information to create a semantic exception explaining the problem in more semantic terms and provide semantic exception information and throw this new exception.
If there is an exception you did not expect, you can log exception information (for example, in a special file which you should never expose), to investigate the problem later.
This can be helpful:
http://www.devshed.com/c/a/PHP/Logging-With-PHP/[
^],
http://www.stanford.edu/dept/its/communications/webservices/wiki/index.php/How_to_perform_error_handling_in_PHP[
^],
http://www.stanford.edu/dept/its/communications/webservices/wiki/index.php/How_to_create_logs_with_PHP[
^],
http://confluence.arizona.edu/confluence/display/WEBPRACTICES/PHP[
^].
—SA