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Well, you’re definitely going to lose your rankings if URLs are changed. Luckily, there’s such thing as 301 SEO redirects that let Google know your pages are being permanently moved to another addresses. As the result, your SEO rankings won’t be affected, even though URL structures will be altered (in your case by adding /product/ part).

If you’re interested in automated migration of your store, I’d recommend you using Cart2Cart. For a moderate additional price, it allows to migrate 301 SEO URLs of categories and products, as well as all your products, customers, orders, review etc automatically, securely and swiftly.

Good luck!

If it was two months earlier, I’d recommend you Bigcommerce. But since they’ve jacked up the prices like nuts lately (and people responded to that -- they’re leaving the platform), Shopify is a non-brainer. It easy to use and has absolutely astonishing, esthetically enjoyable and responsive themes, which is especially important for for a fashion store.

Alternatively, if you’re on a tight budget, you could use WooCommerce too. It’s a bit more complex to set it up properly, but being an open source cart on WordPress, it’s extremely customizable and lightweight. If you have at least basic technical skills, you won’t get disappointed.