Ask to see a neurologist - you have every right to.
Seeing a neurological headache specialist will take a very long time. I waited seven months to see one because I have a separate headache disorder along side 'normal' migraines.
Most neurologists will not put you on a migraine preventer unless you're having migraines almost every single day. Why? The side effects of the preventative medication can be worse and more disabling than the migraines.
I was having migraines at least five or six days out of seven and was eventually put on epilepsy medication to control it. This method of treatment isn't yet licensed by NICE. It took a year to find the right dose to control them though. The side effects are horrible. They include the possibility of kidney stones, it causes extreme fatigue, visual problems, hearing problems, memory problems (so bad at times I have forgotten my address, my phone number, even where I was at that exact moment). I've had periods where I've had diarrhoea and vomiting for days on end, even being unable to keep down water - all in the name of trying to stave off migraines. And it's not foolproof. If I accidentally miss a meal, or if I don't manage to drink 3+ litres of water each day, or go to bed a bit too late then I'll still get a migraine. It became a case of asking myself whether it was worth the trade-off, do I want to sacrifice feeling "okay", to get rid of the pain? I if do sacrifice being "okay" all it will enable me to do is work, I won't be able to have any fun. I do very little else now.
If you're able to take triptans they can be an option for a lot of people. Never take more than two of any type in 24 hours. You need to find one that works for you. I've used so many that none of them work for me. Rizatriptan, Sumatriptan, Zolmitriptan seem to be the most common. They don't just come as tablets either - if you find that tablets don't work fast enough, ask to try an orodispersible (if you haven't already).
But, you can't take medical advice from someone on the internet - you need to find your triggers because you might actually have some and not even realise it. After extensive research it turned out I didn't, I just seem to have vascular problems that need to be controlled medically. And, you need to see a neurologist. Now that I've finally seen two, I have a headache specialist who is helping me deal with my more serious headache disorder and life might actually come together for the first time in years. But again, you need to work with a neurologist to find out what the cause may be and find a solution that suits you.