Protein. More protein. It's normal for one's legs to hurt after a workout, even more so on the second day after. Do you notice that? However it is not normal for your legs to hurt on the bike, or at least not until you have driven them almost to exhaustion. The more usual sensation is for leg pain to go away on the bike due to the production of endorphins.

My rule is that if my legs hurt on the bike, it's a lack of protein in the diet. The amount of protein which is adequate in a sedentary life is not nearly enough if one is exercising hard. Hitting it hard, like intervals, damages the muscles every time you do it. Then they have to repair, which takes protein. So try increasing protein intake a lot over the next few days. Greek yogurt is a cheap and accessible way to do it. Add say 70g of protein to your diet every day, somehow. Dietary protein works pretty quickly. If that's the problem, it should get better within a week. If not, that's not the problem, but it's a quick and easy way to check. Cut back on the fat in your diet to accommodate the extra protein, but don't cut so much on the carbs.

I doubt it has anything to do with fit, tendons, etc. The fact that it's only on the bike when going hard has protein written all over it. If it were other stuff, you'd notice it just walking around. Thus I doubt it's connective tissue. Plus the description "burning" is a muscle sensation. Connective tissue just goes "OUCH". Be that as it may, Berner is correct in his advice. Intervals no more than every 3rd ride. The other two at a pace which causes deep but not rapid breathing.