You might have a problem, I have no speeding tickets, in fact I have not had one since I was 17, but friends of mine that have had as many as you, have not been hired at the places that they applied to, many airlines when they see that many tickets in a short time frame like 7 years, look at it as that you never learned your lesson from your first ticket, I had a friend that had 1 ticket on his record and he got hired, his ticket was 5 years old, and he explanined that he was speeding and he learned his lesson by not getting any other tickets, the way most company's look at this kind of stuff is did you learn from your mistake, can you follow rules and laws, in your case you did not, I would still apply to whom ever you want, if you have a good reason why you got 4 tickets in a 7 year period, then maybe you will not have a problem, but if they have a bunch of people that they interview and you are the only one that sticks out with 4 tickets, you might have a problem but then again, some one may overlook those if you shine in other areas
Good Luck