A dedicated photo-hosting service is usually the simplest fit for this. Based on the suggestions here, SmugMug is a strong option because it supports high-resolution photo hosting and offers API/oEmbed access, and you can also use direct image links in your site markup.

Dropbox can work for small personal collections, but it’s less ideal as public website image hosting. Free usage limits can be restrictive, and some sites or networks may not handle Dropbox-hosted assets as reliably as a service intended for publishing images.

So if your goal is a stable gallery for a website, a photo-focused platform with API or embeddable links is the better recommendation. If you only need something simple for a small personal site, direct links from a storage service may work, but for reliability and presentation, use a dedicated gallery host.