Stapel, I think you have understood the issue perfectly.
Your website has many small files, which each need a request to Apache (the httpd process you see in the stats). To lower the usage, you could have fewer requests to make up each page, which would normally mean having fewer images.
15,000 visitors a day is quite a lot really, as the Shuttle/Voyager plans, although they can cope with a lot, are basically budget-priced webhosting, as they cost upwards from $7.95. I'm not saying that we can't host busy sites, but it really depends on the amount of hits they are getting and the amount of requests they get.
The fact that each image/file is handled individually is the issue. If they weren't, there would be fewer requests and the usage would be lower.
I think that in your case, you could lower the amount of images and stay hosting on your current plan without having to upgrade to a dedicated server. It could well be worth a try, however, upgrading to a dedicated server might be a better option, especially if you think that over the next few years, your usage might go up.
As was mentioned before in this post, bandwidth/disc space and CPU/Memory/database usage aren't the same.
AL3X1S, if I remember correctly, you were using phpBB? Although that is ok, at the time of writing, we recommend SMF instead. In your case, the issue is not really the amount of connections, it is that a PHP script on your account is using too many resources. This could be because the script has been poorly designed, or possibly that your database is corrupted, or just needs optimisation.
phpBB doesn't "prune" posts by default, so the database will grow to be very large. This means queries will take longer, and so will use up more resources. My advice to you is to remove old posts and old users to lower the size of the database, then optimise it. If the usage is still high, make a backup, then convert to SMF.
Although I think that's the main issue on your account, you also mentioned that you were using many "tiny" .gif images. Each of these opens a new request to Apache, so will also raise the usage, however, I think it's a bit of both in this case.
I hope that explains what is happening. If you have any questions, please let us know.