First off, well done you for getting going with your portfolio. It sounds like you're going about everything the right way: building up a body of work online, and writing for websites other than your own.
Before I go too far, I'll say this: you don't need loads of
pieces for your portfolio before you start aiming for paid work. I'd say that three good pieces – those could be blog posts, brochures, sales letters, marketing emails, etc – would be fine. You can then start aiming for paying work, and add those pieces to your portfolio too.
(I'm assuming, here, that you don't need a large portfolio in order to apply for the Master's course.)
One of the best, and easiest, ways to get good clips for a
portfolio is to write guest posts for large blogs. You'll probably want to target blogs that have a fair degree of overlap with the eventual type of writing you want to do (e.g. if you want to do copywriting for small businesses, write for blogs that have an audience of small business owners).
With guest posting, aim high! Choose some of the blogs that you read and love, see whether they accept guest posts (not all will), and write something specifically
for each of them. Go for the biggest blogs first: if they turn the post down, rewrite it and submit it to a slightly smaller blog.
As well as the guest post forming a handy portfolio piece, it may lead clients to you, so make sure your bio includes a brief description of what you do and a link to where they can find out more about your services and hire you.
In terms of copywriting specifically, you may find that you initially need to
work for free, to get some portfolio pieces under your belt. You might look for a non-profit that needs their web copy sprucing up, or ask around in any business-y forums you belong to – you might find someone willing to do a skills swap (e.g. maybe you'll write some sales copy for them and they'll design a logo for you).
It's always worth asking for a testimonial, especially when you're building up your portfolio: you can include that on your website
alongside a screenshot and brief description of the project, with a link to the full blog post / sales page / etc.
The vast majority of clients will be more than happy to give you a testimonial, but very few will think to offer you one unprompted.
Best of luck with the copywriting and the Master's course!
If you have a question about freelancing, or any other type of writing, just reply to this email. I'll
be very glad to address it in a future newsletter or on the Aliventures blog.