I recommend reading the official answer to your question, Appropriate Uses For SQLite. Specifically, the "Situations Where Another RDBMS May Work Better" warns that SQLite does not support concurrent writing:
SQLite supports an unlimited number of simultaneous readers, but it
will only allow one writer at any instant in time. For many
situations, this is not a problem. Each application does its database
work quickly and moves on, and no lock lasts for more than a few dozen
milliseconds. But there are some applications that require more
concurrency, and those applications may need to seek a different
From an appropriateness perspective, I tend to view SQLite as a very sophisticated file format that supports SQL queries. I would tend to avoid SQLite if I wanted to separate my database from my web application, as it is not optimized for this case. In short, SQLite is insufficiently scalable for use in some scenarios, so people running websites who hope to someday become popular may be better off starting with something scalable, rather than going with SQLite and later being forced to switch.
All that being said, SQLite is probably fine for most internal websites; typically internal websites don't require the same level of concurrency and scalability.