  For a fixed finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION
problem asks, given an $n$-vertex input graph $G,$ for the minimum number of
vertices that intersect all minor models in $G$ of the graphs in ${\cal F}$. by
Courcelle Theorem, this problem can be solved in time $f_{{\cal F}}(tw)\cdot
n^{O(1)},$ where $tw$ is the treewidth of $G$, for some function $f_{{\cal F}}$
depending on ${\cal F}$ In a recent series of articles, we have initiated the
programme of optimizing asymptotically the function $f_{{\cal F}}$. Here we
provide an algorithm showing that $f_{{\cal F}}(tw) = 2^{O(tw\cdot \log tw)}$
for every collection ${\cal F}$. Prior to this work, the best known function
$f_{{\cal F}}$ was double-exponential in $tw$. In particular, our algorithm
vastly extends the results of Jansen et al. [SODA 2014] for the particular case
${\cal F}=\{K_5,K_{3,3}\}$ and of Kociumaka and Pilipczuk [Algorithmica 2019]
for graphs of bounded genus, and answers an open problem posed by Cygan et al.
[Inf Comput 2017]. We combine several ingredients such as the machinery of
boundaried graphs in dynamic programming via representatives, the Flat Wall
Theorem, Bidimensionality, the irrelevant vertex technique, treewidth
modulators, and protrusion replacement. Together with our previous results
providing single-exponential algorithms for particular collections ${\cal F}$
[Theor Comput Sci 2020] and general lower bounds [J Comput Syst Sci 2020], our
algorithm yields the following complexity dichotomy when ${\cal F} = \{H\}$
contains a single connected graph $H,$ assuming the Exponential Time
Hypothesis: $f_H(tw)=2^{\Theta(tw)}$ if $H$ is a contraction of the chair or
the banner, and $f_H(tw)=2^{\Theta(tw\cdot \log tw)}$ otherwise.
