"What bothers me is our days and nights are a total mess. [...] I honestly have no idea if it's normal that he eats so often and what to do."
The question has 2 parts:
"[Is it] normal that he eats so often?"
If you count the total number of feedings, you can see how your case fits in the actual distribution of what is observed in real life among different mothers/infants (average = 11 times per 24 hours, standard deviation = 3 times, range = 6 to 18 times). Note that these are actual observations (= what is) of healthy infants. These are not recommendations based on any specific "desired" outcomes (= what should be). Nor are these observations healthy targets. They simply reflect the facts on the ground in one particular study - the actual feeding practice.
It is hard to estimate the number of feedings in your case, because you did not say how long your child sleeps at night (when he feeds less frequently).
"Volume and Frequency of Breastfeedings and Fat Content of Breast Milk Throughout the Day", Jacqueline C. Kent, Leon R. Mitoulas, Mark D. Cregan, Donna T. Ramsay, Dorota A. Doherty, Peter E. Hartmann in Pediatrics Mar 2006, 117 (3) e387-e395; DOI: 10.1542/peds.2005-1417
Infants breastfed 11 +/- 3 times in 24 hours (range: 6–18) [...] Night breastfeedings were common and made an important contribution to the total milk intake. [...] Breastfed infants should be encouraged to feed on demand, day and night, rather than conform to an average that may not be appropriate for the mother-infant dyad. [...] Healthy, exclusively breastfed 1- to 6-month-old infants consume 0 to 240 g of milk between 6 and 18 times during 24 hours, with 64% of infants breastfeeding 1 to 3 times at night. [...] This will provide a normal reference range to enhance clinicians' support for breastfeeding mothers. [...] Data were collected from 71 mothers who were exclusively breastfeeding on demand healthy, term infants who were aged between 1 and 6 months.
"What bothers me is our days and nights are a total mess. [...] I honestly have no idea [...] what to do."
As others, such as @Rekabeka, pointed out, consult a healthcare professional. I would add: in person. They can rule out specific health or feeding issues, beyond the most obvious (weight gain, pee, poop), take into consideration wellbeing of only the infant, but also the parents. Take the advice on this site with a grain of salt: we are not familiar with your specific case.
Gradually space out the feeds. In my personal experience, I and others found this book useful: Suzy Giordano, Lisa Abidin (2006) "Twelve Hours' Sleep by Twelve Weeks Old: A Step-by-Step Plan for Baby Sleep Success". Caveats: (a) This book is not based on clinical studies, (b) I am not a healthcare professional, (c) See the references below for much, much more context and nuance.
Parenting Science blog post: "The best infant feeding schedule: Why babies are better off feeding on cue". It summarizes a lot of research on the topics of feeding schedules and their effects on health outcomes. This blog provides more depth and relevant context to the more narrow and specific statements above. Note that the blog does not suggest spacing out the feeds.
A recent good-quality Cochrane review on the topic provides a different take on this issue (below). The age of infants covers that of the OPs, see references 1-10:
Fallon A, Van der Putten D, Dring C, Moylett EH, Fealy G, Devane D. "Baby-led compared with scheduled (or mixed) breastfeeding for successful breastfeeding." Cochrane Database Syst Rev. 2016 Sep 28;9:CD009067. doi: 10.1002/14651858.CD009067.pub3.
This review demonstrates that there is no evidence from randomised controlled trials evaluating the effect of baby-led compared with
scheduled (or mixed) breastfeeding for successful breastfeeding, for
healthy newborns. [...]
What does this mean?
We looked for studies that compared baby-led with scheduled (or mixed) breastfeeding for successful breastfeeding for healthy newborn
babies. However, no studies were found that met the inclusion
criteria. It is recommended that no changes are made to current
practice guidelines without undertaking robust research, to include
many patterns of breastfeeding and not limited to baby-led and
scheduled breastfeeding. Future exploratory research on baby-led
breastfeeding is also needed that takes the mother's perspective into
FAQs + limitations of my answer:
- Can I assume that "three month old must be right in the middle" [of the distribution, which has the range of 6-18 times per day]?
No. The Kent et al. article simply provides the observed distribution of the number of feedings per day.
- Is 11 feeding per day a healthy target/recommendation?
No. It is the mean (average) of the observed distribution. Note also that this is only one study, and like any other single study, should be treated skeptically.
- The answer seems self-contradictory - why is that?
Evidence in personal experience and in scientific research is often contradictory. Evidence, as shown above, may not always support only one single feeding recommendation for all mothers, all babies of the same age 100% of the time. A responsible answer should discuss competing views, and present more than one-sided evidence (if it exists).